While this does occur, the short answer is ‘usually not.’
Atmospheric air (commonly referred to in clinical settings as ‘room air’) is composed of approximately 78.08% nitrogen, 20.95% oxygen, 0.93% argon, 0.04% carbon dioxide, variable amounts of water and negligible amounts of other gases. Oxygen concentrators use zeolite molecular sieves to separate oxygen from nitrogen in room air and produce a high concentration of oxygen. However, the sieve beds have a similar affinity for argon as they do for oxygen. This results in oxygen concentrators also performing as ‘argon concentrators,’ and the output of both gases increasing by nearly 5-fold in the output from a concentrator (~93-95% oxygen and 5% Argon).
There is no evidence of this concentration of argon (or much higher concentrations) having harmful effects in humans. There are data that argon, similar to some other noble gases (e.g. xenon) may have neuroprotective effects and at very high pressures (beyond seen in the clinical setting) may have some anesthetic properties (PMID: 31470983; PMID: 20472713; PMID: 19500647; PMID: 20236500).
High concentrations of argon will decrease the concentration of oxygen delivered to the patient, and this should be accounted for in the clinical setting. For most clinical settings and scenarios this is unlikely to be clinically significant. This is no different than many of the other gases commonly added to the breathing mixtures (e.g. nitrous oxide, helium, inhaled anesthetics).
The special case of low-flow,  fully or partially closed circuit anesthesia is sometimes cited as a potential scenario where build of argon could occur at higher levels and cause potential harm.  This has been studied and concluded that oxygen concentrators can be used for oxygen supply in fully or partially closed circuit anesthesia delivery systems for indefinite duration as long as the fresh gas flow (FGF) input flow rate is at least twice the oxygen consumption (e.g. FiO2 0.5 and FGF ~1 LPM), nitrous oxide is not used or is being accounted for, and the system is periodically opened for fully closed systems.
Unsurprisingly and unfortunately, arguments citing argon as a reason to not use oxygen concentrators in clinical care are usually driven by those with commercial interest in liquid oxygen.
References:
Parker et al, BJA, 1988