They used to organize pretty decent karaoke nights in Organza Shot Bar on Thursdays and Tuesdays, but lately someone told me that they don't anymore - don't know whether it's true.
Another place with karaoke is Nora (don't know on what days though) on Krakowskie Przedmiescie (right next to the University) - the name of the place (sth between a den and a hovel) pretty much reflects the design, atmosphere and clientele, but if you just wanna get tipsy and scream your lungs out, that's the place.
Also Bolek in Pola Mokotowskie, they used to have a large selection of songs, but that was like 2 years ago, I don't know about now.
I hope somebody here knows some decent and fun karaoke places in Warsaw, cuz I myself would love to find one!