As a mere viewer of The Big Bang Theory, I can't answer this question with certainty, but as a physicist, I can offer a couple observations:
- The equations that appear on the whiteboards in TBBT are from widely varied fields of physics and chemistry. They cover a much wider scope than a typical scientist would actually be working on. Of course, Sheldon is not your typical scientist, and I suppose he could be doing some sort of highly cross-disciplinary research incorporating many different subfields of physics, but still, all the various equations don't seem to have much of a logical connection. Anyway, given that Sheldon considers pretty much anything other than string theory to be beneath him, I can't really see him getting involved in that kind of project on a regular basis.
- Many of the equations I recognize as well-established physics, the kind of thing one learns as a beginning graduate student. This doesn't necessarily mean anything by itself, because all research is based on well-established physics, but if the equations on the boards were meant to reflect actual current research projects, I would expect to be going "huh, what's that?" and looking things up a lot more often than I do. The BBT blog discusses some reasons why it's generally not practical to incorporate actual current research into the show.
- There's one particular scene that may be relevant to this: Leslie is over at the guys' apartment (sleeping with Leonard I think), and overnight she fixes an error in Sheldon's math on the whiteboard which he had been struggling with for some time. The thing is, the particular calculation Sheldon was doing (deriving the QCD beta function from renormalization of gluon-exchange diagrams, IIRC) is something that was first worked out back in the 1970s and has been standard physics ever since. It should have been utterly trivial for Sheldon, as an accomplished string theorist (and genius) to do this properly, so I just have hard time believing that he'd been struggling with it. Even if he were, it would not have been hard for him to just look up the proper method and locate the error that way (perhaps his pride prevented him from doing that, but then again a big part of becoming a successful physicist is being able to look things up).
So in summary, I'd say almost certainly no, the equations do not in general reflect current progress in research (though the latest couple blog entries on the show's blog suggest that they're trying this). But they are at a much higher level than most other instances of math in movies and TV, and they are actual scientific equations, so you have to give the show credit for that.