As a pretty obsessed turkey hunter, I certainly agree with the overall philosophy of killing nest predators whenever they are LEGAL to kill.
With that being said, I have observed some statements in this thread that may be contrary to the law. The only TRAPPING seasons that are open year-round are for beaver, coyote, and groundhogs. The trapping season for all of the other furbearer animals this past year was Nov. 17-Feb. 28. Thus, coon trapping season is currently closed.
Just because coons, skunks, etc eat turkey eggs does not turn them into a nuisance animal where you can trap them anytime you choose. Yes, there is a law that says landowners can destroy animals causing them damage. IMO, it would be a major stretch for a court to rule that a coon destroying turkey nests reached the level of landowner damage. It at least warrants a thorough discussion with the local wildlife officer since they will most likely be the one issuing the citation for illegal trapping.
Again, I'm not against the sentiment of dealing with nest and other turkey predators. However, with the laws in place, it likely isn't a year round deal. It may be something to address with TWRA in the future.