Dont think so, considering two factors: Austria treated as a victim OTl (Anschluss) and i do not think, why it should change and... well, Hungary did not really wanted those territories back. More or less the question compared to any territorial demands pretty much irrevelenat - neglible numbers of hungarians living there, small patch of land, etc.
Not
purely a victim, but certainly closer to a victim than an aggressor. Austria was, after all, divided into formal occupation zones, just as Germany was. But perceptions of it as the first of Hitler's conquests did color that. That would have remained true even if Hungary had not joined the Axis.
But there are other reasons why Burgenland would not be on the table in the latter case. Burgenland was given to Austria after WWI precisely because it was mostly German - it was one more manifestation of Wilson's demand for ethnic self-determination. It wasn't an acquisition by Hitler, like the Sudetenland. In short, it was part of the pre-Hitlerian status quo.
And, as others have pointed out, there was little interest in the territory in Hungary anyway. Nothing like, say, Transylvania.