I don't think it's a felony, but messing with someone else's mailbox is a federal offense. It's definitely illegal to put things in people's mailboxes, as well as illegal to take things out.
It's doubtful charges would actually be brought against someone putting things in a mailbox, though. I think it would be more expensive to hire people to deliver flyers, etc. personally than to mail them, so there's no large scale effort to defraud the government out of their fair share of postage, or whatever the reasoning for the law is.
It's usually some local volunteer inadvertantly violating the law. We ran into this back in Nebraska when my daughter delivered our suburban town's small weekly newspaper and the military base's weekly newspaper - for a couple of weeks, she thought putting the newspaper in the mailboxes where they'd stay dry would be a good idea. Putting the base newspaper in the mailboxes was kind of ironic in that she managed to make the government an accomplice in violating federal law. Of course, being a kid, she was just told she had to stop.