This one was fun to look into, as I've often wondered about those bars, as well as football players/other athletes as they sit on the sidelines and suck down O2 from a mask.
Short answer: Nope!
That said, for a "normal, healthy individual without respiratory problems," they're not really gonna hurt anyone, either.
Long answer, with science and stuff:
From a small study that was done in 2004 at Indiana University, researchers took some college students with no heart or lung conditions, and did a single-blind test where some were given oxygen to consume, and some were given regular compressed air. They found that (emphasis mine):
Data gathered from the evaluation concluded that there was little to no difference with SaO2, HR, and post-VAS measures, or improvement post-gas administration between the oxygen and sham group. Calculations indicated there were no statistical significances between those who received oxygen and those who received air. Any purported beneficial effects by proponents of oxygen bars are yet to be substantiated. Large scale studies determining the benefits of oxygen bars should be conducted.
Check out the abstract there for a table with the actual figures. The table goes a long way towards showing that people claiming to physically feel different are probably heavily placebo influenced. Their heart rates are elevated, but the oxygen content of their blood is the same (more on this later). It follows that some people might feel (un)pleasantly lightheaded or something else, but it's unlikely to be anything serious or particularly meaningful.
Earlier than that, a report from a 2002 issue of the FDA Consumer (reproduced here at a different website) about oxygen bars made a lot of points to help debunk the idea that inhaling extra oxygen is good for you/useful at all (emphasis mine):
The American Lung Association says that inhaling oxygen at oxygen bars is unlikely to have a beneficial physiological effect, but adds "there is no evidence that oxygen at the low flow levels used in bars can be dangerous to a normal person's health."
Comparing to what we normally breathe (emphasis mine):
But there are no long-term, well-controlled scientific studies that support these claims for oxygen in healthy people. And people with healthy lungs don't need additional oxygen, says Mary Purucker, M.D., Ph.D., a pulmonary specialist in CDER. "We've evolved for millions of years in an atmosphere of about 21 percent oxygen."
Concerning how the bars generally get the oxygen to you, and its composition (emphasis mine):
Many oxygen bars use a concentrator, which filters out the nitrogen and other gases in the air circulating in the room, and then delivers the concentrated oxygen, about 95 percent pure, through a hose at a continuous flow rate. But oxygen users inhale the surrounding air along with the oxygen pumped through the nose hose, which decreases the concentration. The concentration is further decreased when oxygen is pumped through an aroma. According to one oxygen bar supplier, the customer gets less than 50 percent pure oxygen.
Although breathing these low levels of oxygen may not hurt a healthy person, "people have nothing to gain by frequenting oxygen bars, and subject themselves to unnecessary risk," says Purucker.
As I alluded to above, and as Dr. Purucker explains, humans are pretty heavily adapted to optimally breathe oxygen from an en environment of ~78% nitrogen and ~21% oxygen. In fact, blood leaving the lungs is already 97% saturated with oxygen, just from breathing normally. It seems highly unlikely that even breathing a solution of 95%-100% oxygen is going to do much more to optimize one's blood oxygenation.
Finally, I know it wasn't explicitly asked about, but I feel like it's also worth including some information about athletes breathing supplemental oxygen. The same FDA report explains about athletes that:
"They don't need it," says Conrad Earnest, Ph.D., director of exercise physiology at the Cooper Institute in Dallas. "It's one of the biggest placebo effects going," he adds. "It's a combative activity, so yes, the [football] players are going to be out of breath, but it's because of massive exertion--not because of lack of oxygen." The exception, says Earnest, might be athletes who play at higher elevations than they are used to, and don't have time to acclimate.
And an older study (subscription only, sorry) from the Journal of the American Medical Association found that (footnotes mine):
[Winter et al's]1 simple, but elegant, experiment shows no difference in plasma lactate levels after a maximal exercise test, or in subsequent maximal exercise performance after breathing 100% oxygen for 4 minutes of recovery. Strengths of the study include a randomized, double-blind, crossover design and the study subjects were highly trained... professional soccer players.
1: Winter FD, Snell PG, Stray-GundersenJ. Effects of 100% oxygen on performance of professional soccer players. JAMA. 1989;262:227-229
It goes on to reiterate what was already concluded here, this time by citing a study over 30 years ago (footnote numbering theirs, source and emphasis mine):
The nearly complete saturation of arterial blood with oxygen while breathing atmospheric air makes it improbable that oxygen enriched air will have any benefit on recovery from exercise. Moreover, although oxygen is usually administered to athletes with the belief that it will aid in lactate clearance, it is ironic that athletes would probably lower lactate concentrations more rapidly if they would walk during recovery rather than sit and breathe 100% oxygen. In 1975, Beicastro and Bonen7 showed that lactate removal from the blood was more rapid if subjects exercised at 30% or 45% of VO2max than if they rested immediately after strenuous exercise.
7: Belcastro AN, Bonen A. Lactic acid removal rates during controlled and uncontrolled recovery exercise. J Appl Physiol. 1975;39:932-936.