Well, I imagine it doesn't end well for the monkey, but does the racer have to stop to pick up his dropped monkey? DQ'd? or
Well, the rules are that both you and your monkey have to cross the finish line together for it to count, going by the testimonial of one who got to keep their 8th place finish because, while he'd fallen off the bike just before the finish line, both him and the bike crossed in the #8 spot. Otherwise it would have been a DNC.
So, assuming the monkey isn't too bruised up to continue the race, the racer would indeed have to stop to pick up the passenger. Or maybe circle around with the lap not counting. Monkey calls it quits, Did Not Complete.