Gilligan's Island, Petticoat Junction, Green Acres, The Beverly Hillbillies all mentioned either the series name or lead character in the theme song lyrics. The last television series that I can recall that did this was The Love Boat (1977-1987). Cam anyone think of a television theme song after 1987 that did this? The Friends and Cheers theme songs were

connected to their series, but the series and characters are never mentioned in the lyrics.

The Nanny, with Fran Drescher

Crazy Ex-Girlfriend

Fresh Prince of Bel Air

The theme song to Frasier did - the last line was "frasier has left the building"

The only one I can think of is South Park.

I think we have at least one winner. South Park is still on the air.

All of these answers just go to show why television is one of my worst trivia categories

Crazy Ex-Girlfriend may not still be getting new episodes, but the theme song is way newer than South Park's, as is Unbreakable Kimmy Schmidt's:
Unbreakable Kimmy Schmidt | Opening Theme by Jeff Ri...

And New Girl's:
New Girl - Opening

The Orange Is the New Black theme sounds more like a radio song adapted for the show's titles, but it was actually written specifically for the show (and then expanded to have a radio version as well):
Orange is the New Black | Opening Credits | Netflix

As far as I can tell, the same is true of the theme song to Community. In my five minutes of googling, I didn't find if it was released prior to the show if it was written for it.
NBC's Community Opening Theme Song

Grey's Anatomy now uses an instrumental version of its theme song, but for the first couple of seasons, it had lyrics.

We're getting a little older now, but Firefly:
Firefly Opening Theme

And less old, Sons of Anarchy:
Sons of Anarchy: Complete Intro/Opening Credits (All...

So theme songs with lyrics are definitely not as common as they used to be, and usually they're written as catchy pop songs rather than as "here's what the show is about," though there are a few exceptions (Crazy Ex-Girlfriend, New Girl, Kimmy Schmidt).