There is no hard and fast rule on this. My years of OTB play has shown me that 99.9% of the time a draw is agreed to. In those rare cases where someone has a lot more time, then they will keep playing for the win.
Rook vs Rook and Time
Thanks, but i've had this happen several times now so the 99.9% certainly doesn't seem to apply to the players on here - who must win on time regardless if it take 100 moves.

Yes, if you're playing online blitz games with no increment then don't be surprised if people decline the draw offer and keep playing for a win. OTB, especially in longer games with increment, it is very common for a draw to be agreed upon immediately as there is no chance for either player to win on time.
I just played someone and we got down to Rook vs. Rook. I obviously offered draw as we went in circles a few moves. He would not accept. He was up about a minute in time and ran clock out on me instead of draw. Is this professional? And how would this be handled in a tournament? Seems like this could happen often and require lots and lots and lots of moves to run the clock out and rather odd with just a rook each. Sorry as this probably has been asked before but I couldn't find.