Q:

Will homemade facemasks using vacuum bag filters cause lung irritation? 

In this YouTube video A Doctor Explains How to Make the Safest Face Mask, a doctor, and then his wife, talk about 'the safest' home-made face mask. 
(He disclaims that it has been tested, but says he is confident it is safe.) 
This mask was made with a HEPA filter vacuum-bag, with no fabric liner. 
The making process starts at 2:34 minute. 
The vacuum bag brand that they use is this one:

HEPA, genuine parts, Hoover, traps 99.97% dust and pollens down to 0.3 microns. 

There was some controversies in the comment section (not quoting because there was no mentioned references). 
Primal Survivor warns against mask designs like this:

Important:
Don’t wear a HEPA vacuum bag directly against your skin. 
HEPA vacuum bags have fiberglass threads inside of them.  The fibers are large so aren’t dangerous like fiberglass dust.   However, you could still breathe them in, which could cause airway irritation. 

...

Instead, you must use the HEPA vacuum bag as a filter inside a cloth mask. 

Will a facemask made of a HEPA filter, without a lining, cause lung irritation? 

A:

Yes.  They can.  And many if not most or the vast majority will.  Technically and practically they are not very suitable to begin with. 
That is yes for many brands out there and yes for many possible applications if a homemade mask is made out of them.  Because there are so many possible combinations out there, finding one such bag is difficult if that bag should not be inherently dangerous if used for this purpose and made and worn according to the necessary precautions, 
Numerous warnings list numerous reasons for why this is a terrible idea in general. 
What all these disadvantages have in common:

These vacuum cleaner bags are not designed to be used in such a way. 
It is unclear what manufacturer uses which material. 
'False sense of security' may lead to risk compensation. 
While Virions may be kept back to a large but far from perfect degree from airflow through the material, the thick cloth will facilitate side stream airflow if not fitted really well, presenting a large portion of inflow that is not filtered at all. 
The pressure difference a vacuum cleaner generates is far greater than what humans lungs can sustain.  Breathing will be a distress factor, and will become shallow and faster, with negative consequences for blood pressure, heart and lungs.  Shallow breathing is a risk factor for inadequate oxygenation and pneumonia. 

Point 4 and 5 present a target dilemma.  
In addition to point 5: It seems self-evident that vacuum cleaner derived filtering facemasks (vdcff) are less suitable than either surgical masks or N95 masks, while vcdff are much more difficult to breath through than either of the first.  Yet even N95 and surgical masks are so obstructive for the airways that they can induce headaches and hypercapnia, the latter immediately after putting on the masks.* 
 Always wearing such a professional mask is not advisable, and often not even possible.  Vcdff are much worse for that as a matter of principle. 
The question of do "vacuum cleaning bags contain fibreglass" may be difficult to ascertain in this generalised way.  Some do, and yet:

Many manufacturers are preparing statements to show their vacuum bags do not contain fiberglass, …

And instead have a woven or more often shrink or melt blown material of 100% polypropylene or other artificial fibres.  
Warnings from manufacturers and sellers issued specifically against any intended use as face mask include the following:

Due to the Coronavirus pandemic, Shop-Vac Corporation has received numerous inquiries about filtration materials in its products.   
This communication is meant to answer such inquiries. 
First, the filter materials used in the vacuum cleaners are for no purpose other than to protect the end users of Shop Vac products from particles or debris being expelled back into the ambient air during operation.   They are in no way designed or intended to protect humans from bacteria, viruses or other pathogens.   Second, direct contact or coverage of the human mouth or nose with the filter materials are strictly forbidden for any purpose.   Under no circumstances should anyone attempt to make a mask or mask material from any filters manufactured, sold or distributed in the market for or by Shop-Vac Corporation.   These materials were not designed, intended or designated for this type of use.   Please consult the CDC website for more information at https://www.cdc.gov/coronavirus/2019-ncov/index.html. 

It's worth remembering that the airflow in a cleaner is exactly the other way around than how it would be worn as a makeshift face mask.  The interior of these bags may be coated with all kinds of stuff.  And if that is then pressed against skin or sucked through, these coatings may be absorbed into the skin or inhaled into the lungs.  Not good. 
How these bags are constructed is up to the manufacturer. " HEPA" is now a generic term that just says what a product promises to do, not what it does or how it does this, nor with which materials it wants to achieve that. 
That fibreglass is a suitable material in principle is evidenced by looking at some types.  This non-washable replacement filter is for example called "360° Glass HEPA and Activated Carbon Filter".  These do not have to come in such a rigid form. 

Terms to look for when buying an air filter cartridge
  Glass, Glass Fibers, Glass Microfiber, Spun Glass, Fiberglass

If a bag contains spun glass then breathing directly through the material is already bad.  But as a bag should not give much of this material into the environment anyway, this looks 'safe' at first. 
But to make a mask you need to cut through the material, producing many tiny strands and fragments, ready to be inhaled.  And for that or skin irritation we need to consider a quick Fiberglass factsheet. 
And the catch-22 irony here is that if such a mask gives off tiny glass fibre filaments, then to catch these you need another HEPA filter, or an N95 mask, and not just some jersey cloth…
That is of course just the start.  Such a bag may be entirely plastic-weave, polyester microfiber or whatnot.  Inhaling micro plastics will do no good to lung alveola.  So a panic stricken focus on fibreglass is already overspecific, since the alternatives are coming with their own set of 'drawbacks':

The annual production of plastic textile fibers has increased by more than 6% per year, reaching 60 million metric tons, about 16% of world plastic production.  The degradation of these fibers produces fibrous microplastics (MPs).  Such MPs have been observed in atmospheric fallouts, as well as in indoor and outdoor environments.  Some fibrous MPs may be inhaled.  Most of them are likely to be subjected to mucociliary clearance; however, some may persist in the lung causing localized biological responses, including inflammation, especially in individuals with compromised clearance mechanisms.  Associated contaminants such as Polycyclic Aromatic Hydrocarbons (PAHs) could desorb and lead to genotoxicity while the plastic itself and its additives (dyes, plasticizers) could lead to health effects including reproductive toxicity, carcinogenicity and mutagenicity. 
— Johnny Gasperi: "Microplastics in air: Are we breathing it in?",  Current Opinion in Environmental Science & Health
  Volume 1, February 2018, p1–5.  10.1016/j.coesh.2017.10.002

The manufacturer of the product pictured in the question keeps the exact construction materials for this product well hidden.  But the following is found at the product page:

WARNING: Cancer and Reproductive Harm - www.P65Warnings.ca.gov 

Since the bag appears to be of a triple layer design with an external brand-name as 'the HEPA' part, and that external brand name advertises some of the products under that brand as "outperforms fibreglass", the hazardous nature of this product is probably not from that HEPA layer. 

High-filtration microfiber with 3 layers of Filtrete HEPA media

A sweeping statement of "bags do never contain fibreglass" seems out of place:

many vacuum cleaner HEPA filters are manufactured with fiberglass. […]  
[…] please make sure to do a lot of research!  It is very important to make sure the filtration media you purchase does NOT have fiberglass, as some do.  You do not want to breathe in Fiberglass.  

Sometimes this is a desired feature to be advertised, random market place:

Alibaba.com offers 325 fiberglass filter bag for dust collector products.  About 83% of these are Filter Bags

As explained here:

What Does a HEPA Filter do? 
  HEPA stands for high-efficiency particulate air.  HEPA filters are extremely effective at taking micro-particles out of the air.  It is these micro-particles that cause pollution.  The filter itself is formed of fiberglass fibers that are randomly arranged to trap large and small particles.  HEPA vacuums are able to trap even the smallest pollutants because the air is forced through the fine mesh and any particles will cling to the fibers within the filter. 

Additionally these bags may be constructed in layers.  Different layers with different purposes. 
The drug stores put out warnings like these:

"Products developed exclusively for vacuum cleaners can be harmful to health," warns Kerstin Erbe from the management of dm. 
  Antibacterial powder is harmful to health
Many vacuum cleaner bags contain a fine antibacterial powder that can be released by cutting the bags open. " If you now use the vacuum cleaner bag as a mask directly in front of your mouth and nose, this powder is inhaled and is harmful to both lungs and digestive organs. 

A polymer powder full of 'chemicals', i.e. antibacterial agents and sometimes fragrance, or really just everything a manufacturer can think of. 
Another seller warns:

Contrary to the increasing number of tips on the Internet, we at the [manufacturer] cannot recommend using [our] coffee filters or [our] vacuum cleaner bags as homemade face masks to protect the mouth and nose. 
[Our] coffee filters have fine pores that filter particles of a certain size.  Corona viruses are many times smaller than this, with a size of 0.12 - 0.16 µm, and are therefore not completely filtered.  Coffee filters do not provide the protection required. 
The same is true for [our] vacuum cleaner bags.  They have a fine dust filter which filters 99.9% of fine dust down to a size of 0.3 µm.  However, corona viruses are smaller with a size of 0.12 – 0.16 µm.  Therefore, they do not provide the protection required from a face mask. 
We would also like to stress that home-made protective masks do not necessarily fit as well as masks that you can buy.  If a mask does not fit correctly then it will not provide the best protection. 
Another problem is that people may feel safer with the homemade masks and therefore be more careless when it comes to other important hygiene recommendations, such as social distancing, washing hands, etc.  An ill-fitting mask could even increase the risk of people becoming infected as it may lead them to touch their face more often and become infected in this way.  
Where required, professional protective masks should be used. […] 
This non-woven material is designed to meet the requirements necessary for use in a protective face mask.  The material is used by specialised companies to professionally and efficiently manufacture protective masks, which are then delivered to medical institutions such as hospitals and care homes, etc. 

The German Army University published a study that in principle only looked at filtration efficiency, but after concerns over internet-pundits gobbling up the  results as 'proof' a later version now explicitly lists:

We strongly recommend using the outside of the dust bag as the inside of the respirator for the following reason.  In order to obtain approval, the vacuum cleaner bags must not release any dust or parts of the bag itself (fleece, glass fibre, synthetic fibres or other filter materials and additives etc.)  outside into the ambient air.  Since the users of the mask do not want to inhale any of these substances either, we strongly recommend using the outside of the bag as the inside of the mask. 
– Christian J. Kähler, Rainer Hain: "Strömungsanalysen zur SARS-CoV-2 Schutzmaskendebatte.  Über Abstandsregeln, Mund-Nase-Schutz, partikelfiltrierenden Atemschutz, Filtermaterialien und Maskenfertigung", Universität der Bundeswehr München, Institut für Strömungsmechanik und Aerodynamik, 11.04.2020. ( PDF)

Finally, a study factoring in the actual humans breathing through the mask:

The surgical mask had the highest filtration efficiency when challenged with bacteriophage MS2, followed by the vacuum cleaner bag, but the bag’s stiffness and thickness created a high pressure drop across the material, rendering it unsuitable for a face mask.  Similarly, the tea towel, which is a strong fabric with a thick weave, showed relatively high filtration efficiency with both B atrophaeus and bacteriophage MS2, but a high pressure drop was also measured. 
Comfort should be an important factor in the material used to make a homemade mask.  The pressure drop across a mask is a useful measure both of resistance to breathing and the potential for bypass of air around the filter seal.  If respiratory protection is not capable of accommodating the breathing demands of the wearer, then the device will impose an extra breathing load on the wearer, which is especially impractic- able for people with breathing difficulties.  Furthermore, the extra breathing load may induce leakage owing to the increased negative pressure in the face mask. 
— Anna Davies, Katy­Anne Thompson, Karthika Giri, George Kafatos, Jimmy Walker and Allan Bennett: "Testing the Efficacy of Homemade Masks: Would They Protect in an Influenza Pandemic?",  Disaster Medicine and Public Health Preparedness, Available on CJO 2013 doi:10.1017/dmp.2013.43

The expert advice on this:

According to Gabriele Ettenberger-Bornberg, expert for hygiene, cleaning and disinfection at the Austrian Research and Testing Institute (OFI), vacuum cleaners contain so-called HEPA filters in most cases.  These are also questionable. " These consist of glass fibres, others are made of polymers.  If you inhale these glass fibres, they can get into your lungs and have harmful effects on your health," she tells futurezone.  
According to the expert, it is impossible to say whether there are any vacuum cleaner filters that are harmless as protective masks. " These filters are not tested for inhalation, but rather for their ability to retain bacteria and the like.  They are not a medical device - there was previously no need to test them for health effects," says the researcher. 
She advises to use medical devices that have been tested for biocompatibility and are skin-friendly.  

*
Acta Neurol Scand 2006: 113: 199–202 DOI: 10.1111/j.1600-0404.2005.00560.x 
"Headaches and the N95 face-mask amongst healthcare providers"
"Effects of long-duration wearing of N95 respirator and surgical facemask: a pilot study", Journal of Lung, Pulmonary & Respiratory Research, Volume 1 Issue 4 - 2014. 
Ulrike Butz: "Rückatmung von Kohlendioxid bei Verwendung von Operationsmasken als hygienischer Mundschutz an medizinischem Fachpersonal", Dissertation, 2004. 
Should the question relate to the current global event, or be read as such, this should be required reading: Respiratory virus shedding in exhaled breath and efficacy of face masks.