While I am not sure if it was necessarily "controversial," Mr. Rogers did testify in front the government twice: first in 1969 to Congress, to respond against proposed budget cuts to PBS and the Corporation for Public Broadcasting, and the second to the Supreme Court in 1979 to support home-recording of television shows (especially for the purpose of allowing families to watch them together at a later time). In the 1979 case, his testimony was actually cited in the ruling.

A lot of people tend to bring up some military past, accusations of child molestation, tattoo sleeves, as so on and so forth. I don't know that Snopes is necessarily the most reliable website, but their breakdown of all these rumors is pretty damn comprehensive.

Something worth mentioning is that humans have a habit of raising certain individuals to a god-like status, so any negative representations of Mr. Rogers could have just be suppressed or willfully forgotten.

By all accounts, though, Fred Rogers did not play a character on TV; we were just fortunate enough that cameras happened to be pointed at him so often.