The idea of involving real money in SE activity has been discussed a few times on meta sites (e.g. 1, 2, 3, 4, 5 as posted by AccidentalFourierTransform in a comment), and while there seems to be broad agreement that it would be a bad idea, I don't know of any rule that explicitly and directly forbids it. Assuming such a rule doesn't exist, I think part of the reason it doesn't exist is that people recognize it'd be impossible to enforce. The moderators and SE staff don't generally know (or care) what you do outside of the network.
Personally, I think if you offer anything more than a trivial amount of real money for bounties, it could easily lead to a situation where people are putting bounties on questions that don't deserve them and giving you an excessive share of the attention that comes to featured questions. That would be bad for the site. So I'd suggest that you not do it. But that's just me, speaking unofficially, that's not an outright rule as far as I know.