They have tons of contests, but I don't think they have too many eating/drinking contests.
However, they do have Fun Ship Fear Factor. Here is how it works (or how it worked on my cruise, at least) - they select a few volunteers, you have to do three separate contests, and they eliminate a few people after each round. For us, round 1 was dancing in front of everyone. Round 2 was popping ballons without using your hands ("fear of latex"), and the last one was a blindfolded eating contest. I got a big bowl of what I could only guess was a mixture of rice, grapes and spicy curry (for liability reaons, I don't think they could do any of the really nasty stuff they do on the real Fear Factor).
I'm proud to say that I won, got a trophy for it, and was featured on the cruise video looking like a moron