Are there any travel websites that offer incentives for users to achieve certain tasks on their travels?

This probably shouldn't be an answer, but it's going to be too long for a comment, so I'll put it here. Feel free to convert if you really don't like it here.

Providing tangible rewards for 'crowdsource' feedback hits two major obstacles which have not yet been solved.

1. Verification. We already have very little way of determining if a review of a restaurant, hotel, or place was written by someone who actually went there. We just have to take their word for it. The only reason to believe them is "why would they bother?" i.e. who would bother to create a plausible-sounding review of a place they hadn't been to? As soon as you reward people, that changes. I'm pretty sure I could write 100 plausible-sounding reviews in a day if I thought it was going to get me a free vacation. (N.B voting doesn't help with this. If you go to the trouble of writing a fake review, then creating 25 fake accounts to vote it up is no extra trouble)
2. Bias If an establishment is going to give out free stays for reviews, you'd better bet it's not going to give them out to reviewers who trash them. That's going to really cut down on the number of negative reviews, as reviewers compete to be as nice as possible and get their free stay.

If course both of these are problems now to some extent. And sometime in the future the internet may solve the problem. But not yet.

EDIT: Since writing this seven years ago it has of course turned out to be a major problem. Travel organizations and others are now routinely paying people to write favourable reviews on websites.