Q:

Would hydrogen fuel tanks on an airplane increase lift? 

Since hydrogen is lighter than air, would hydrogen fuel tanks help provide lift for the airplane? 

A:

Let's do some quick math.  Air has a density of about 1.3 g/L and hydrogen has a density of about 0.09 g/L. The best case for lifting force would be storing the hydrogen at the same pressure as the surrounding air.  This means that every L of hydrogen can displace 1.21 g of air, reducing the density of the aircraft.  The amount of air displaced by the aircraft (and therefore the buoyancy) would not change, but the density of the aircraft in that space would decrease.  This would decrease the lift needed, but let's see by how much. 
The greatest amount of free volume in an aircraft (that isn't part of the pressurized section of the fuselage) is the fuel tanks.  An A380 has a fuel capacity of about 323,000 L. If we consider filling half of that with hydrogen, this would result in a total reduction of about 190 kg.  An A380 has a max takeoff weight of 575,000 kg.  This change would help but would be very small compared to the total lift needed. 
As David Richerby points out, hydrogen needed as fuel would need to be pressurized (if not liquified) to fit enough onboard.  This would increase the density above that of any displaced air, which would instead add weight to the aircraft. 
Perhaps the hydrogen could be produced instead of stored.  Newer aircraft do have something called an OBIGGS, or onboard inert gas generating system.  This system generates nitrogen which is used to fill the space in fuel tanks after the fuel is used.  Being an inert gas, nitrogen can lower the amount of oxygen in the space below what would be required for ignition of the fuel vapors. 
However, this system splits nitrogen out of air, considering that our atmosphere is mostly nitrogen.  Hydrogen would have to come from somewhere else, usually water, which would add more weight.  Another problem with hydrogen is that when added to air it can be explosive all by itself, which would make the situation even worse.  Considering the minor benefit, this would not be a system worth installing on an airliner. 
Regarding the feasibility of a hydrogen powered aircraft, NASA did a study and decided that this is not a good source of power for an aircraft.  The volume and weight required to store even liquid hydrogen would be too much for the aircraft to be very useful. 

A:

No.  If you want to use hydrogen as fuel, you need a lot of it, which means you need to pressurize it.  Hydrogen is less dense than air when it's at ordinary atmospheric pressure but, as you start to pressurize it (i.e., fit a greater mass of hydrogen into a fixed volume), that advantage decreases.  Worse, if your fuel is pressurized, you need to store it in a strong tank and strong means heavy. 

A:

Buoyancy (a.k.a.  static lift) depends on the volume of the object.  If you put something else in the tanks, the buoyancy does not change. 
What can change, however, is the weight.  If you wanted to use it as fuel, you'd have to take such amount as to get the same combustion heat as you get from kerosene.  Hydrogen has lower heating value 119.96 KJ/kg and kerosene has ~43 MJ/kg, so you would need almost three (2.79) times less hydrogen by weight.  This is totally unrelated to its density, but it is the only way how it can offer advantage as fuel. 
However:

You'd still need to squeeze a lot of it into the aircraft, which means either huge pressure or very cold. 
Even liquid hydrogen has density of only 70.85 kg/m³ (gaseous at 300 bar has only 26.96 kg/m³) compared to 800 kg/m³ for kerosene, so even for three times less weight you'd still need four times larger fuel tanks and they would come with higher form drag and higher weight.  They would also have slightly higher buoyancy, but the drag penalty would be higher than any gain from that. 
Dealing with stuff colder than −253 °C is complicated and it takes a lot of energy to keep it cool it and keep it cold.  For spacecrafts the weight advantage is so important that its worth it, but for aircraft it isn't. 
Compressed hydrogen is not really an option.  The tanks would be heavier than what you could save in the weight of fuel. 

So overall hydrogen does not have any advantage to offer for aircraft.  It only has it for spacecraft that leave the atmosphere quickly so they don't care about aerodynamic drag much, but care about weight a lot.