I'm not convinced that this is a Dropbox issue. I am having exactly the same problem, but my watched folders are not in Dropbox (although I do have Dropbox installed). Renaming the folder does not fix it. But removing and re-adding rules to the folder in Hazel does seem to fix it (at least for a while).
Seems like Hazel is being de-registered for fsevents on that folder, and the act of changing the ruleset will cause Hazel to re-register itself for fsevents on that folder?
Also seems like Hazel does also run rules on a timer, as a fallback. Perhaps Hazel should always re-register itself for fsevents on watched folders when it runs on a timer, as a work-around, if we cannot determine exactly why it is becoming de-registered to begin with.
See:
viewtopic.php?f=4&t=2372&p=12286#p12286