A lot of it depends on the scale of company that you're working with, whether a union is involved, and how people's relationship's stand. In my very first job out of college, I was an ASM at a union theatre. There were very strict rules regarding what I could do verses what the IA hands took care of. I slipped a few times, but the guys were fairly cool about it since it was small stuff and they knew I was new. There was a situation where everyone was comfortable with a 'the show has highest priority' mentality, though I definitely couldn't constantly break the rules.
As its been said, it all depends on a case by case scenario - I wouldn't get into the habit of thinking of it as okay, its only something to do in serious circumstances. Even if everyone is getting along fine now, if you step outside of your department too many times, that can give you a bad rep as interfering - even if you're only doing it for the show's good. Lets say you hemmed a frayed handkerchief - do you know for a fact that was the final prop? That wasn't something that was borrowed, that they thought would be final, but it doesn't quite look right, so they are supposed to return it untouched? If they're really busy, they might forget to say that its not going to be the final prop after all.
This was just a hypothesis - there are times when you can use your own judgement as far as taking care of something, but really, I'd check with your prop department before you did anything - just to say, heh, I know you're behind, I'd like to help out, do you mind if I take care of this item for you?
-Centaura