When people say this I always think, well surely they are both the same. Either put up with it or shut up. No difference so where does the saying come from?
To 'put up' in this situation, means to stake...ie to lay a bet, as it were. Thus, 'put up or shut up' means exactly the same as 'put your money where your mouth is'. So, these two instructions suggest you should back your claim with hard cash if you're so sure of yourself.
quizmaster is right, i think it came from poker games while bluffing, ethier put up your money, or shut up! this of course is when "table talk" wasn't cheating and it was just the players "egging" each other on, or talking smack!