I feel like this should be thing. What you do is pick a song, that Weird Al has parodied, and sing Al’s lyrics instead of the original’s to see if anyone in the audience notice. Obviously it would be somewhat challenging as you’d have to memorize the parody lyrics and ignore the ones on the screen. I started thinking about it as I was listening to his version of Miley Cyrus’ “Party in the USA” called “Party in the CIA”. The original’s lyrics are pretty lame but the parody is funny.
I haven’t sung karaoke in 15 years or so. But one time, I did sing the lyrics to Weird AL’s “Headline News” to “Mmm Mmm Mmm Mmm” by the Crash Test Dummies.
Not the same, but sometimes when I sing Gloria Gaynor’s “I Will Survive,” I sing it like the CAKE cover of the song.
I’ve sung “You’re Pitiful” over the karaoke track of “You’re Beautiful”. I don’t think anybody but the KJ really got it. He thought it was pretty funny though.
I’ve also Richard Cheesed it on “Copacabana” a few times.
Sometimes I perform karaoke mashups of (mostly) my own composition. The best is when the audience reactions go from initial pleasure at what they think is the original song, to bewilderment, to realization and appreciation. Often it doesn’t get past the bewilderment stage, but that is satisfying in its own way.
Almost. My family and I were at one of those piano duels and the players played American Pie. My son and I sang The Saga Begins, which pissed off my other kids.
I have several Allan Sherman albums, and some stuff by Homer & Jethro. And listening to Dr. Demento for several years honed my skill at parody lyrics–including Jeff Weinstein’s rendering of “The Girl from Ipanema” to comment on Zsa Zsa Gabor! 
I’ve Tom Jonesed it to Prince before.
I shoot concert videos by the School of Rock, and at one show they were covering “Total Eclipse of the Heart”. But the kids had seen the “literal” version, where someone re-wrote the lyrics to reflect what is going on in the completely deranged music video. They considered doing that version instead, but concluded that it didn’t work without the video.