The problem with this is that it seems likely to be biased against unexpected results or results that contradict the dominant theory. The old saying, "Science advances one funeral at a time," has a lot of truth in it.
[Edited as enlightenment dawns ...]
Science advances one funeral at a time. The stock market accelerates the process by separating fools from their money.
This betting market only covers reproducibility, not truth. But since most reproducible findings are still false, betting contrary to bias is unlikely to work.
Toy example to illustrate my claim: Suppose a given finding has a probability p of replicating, but the biased market estimates q < p. This means that you must spend $q and if the experiment replicates you'll earn $1.00.
On average you'll win $p from these bets - $1.00 exactly p of the time. Your net winnings will be $p-$q. As long as your theory more accurately predicts the replication probability, you make money.
At least in the context of the phrase 'science advances on funeral at a time', usually folks are talking about studies that are wrong for reasons other than replicability. (Because in general it's pretty easy to convince folks that something is false if it can't be replicated, but much more difficult to convince folks that there is some deeper methodological or epistemological issue.)
I.e., observe a bernoulli trial and see 60 successes in 100 trials. Plan a followup study which will repeat the bernoulli trial 100x - predictions would be on the # of successes in the followup, not some statistical analysis.
The phrase "science advances one funeral at a time" is somewhat orthogonal to this idea - it tends to be about older scientists being unwilling to accept new theories in spite of repeated successes. Such scientists would lose their money if they bet or be ignored if they didn't.