The show's basic premise is a riff on Brecht/Weill's The Rise and Fall of the City of Mahagonny (an unnamed city on the brink of catastrophe), crossed with The Cradle Will Rock (the poor downtrodden rising up against the evils of capitalism).
The character names in Urinetown also are a riff on the character names in The Cradle Will Rock. Mr. Mister / Mr. Cladwell ... Larry Foreman / Bobby Strong.
Musically, the show starts off with songs that bear either musical, thematic or structural similarities to the score of The Threepenny Opera. The Overture is almost a note for note homage to the Overture to Threepenny Opera. "Urinetown" resembles "Moritat". "Privilege to Pee" is "Peachum's Morning Prayer". "Cop Song" is "Cannon Song".
"Follow Your Heart" does resemble the balcony scene from West Side Story, but musically, sounds more like an homage to the songs of Marc Blitzstein.
After that point, the satirical base broadens, and people claimed all sorts of different influences, from Les Miserables (Act One Finale), Fiddler on the Roof ("What is Urinetown?"), West Side Story (the "Cool" references in "Snuff that Girl"), Guys & Dolls ("Run Freedom Run" is in the Broadway faux gospel mode of "Sit Down You're Rocking the Boat").
John Carrafa's musical staging went after specific Fosse and Robbins trademark moments, but also included some cheeky references to Alvin Ailey's "Revelations" in two different spots.
Other songs satirize Broadway show writing in general, more than any specific song. "Mr. Cladwell" strikes me as designed to resemble any hat and cane number for a leading man. "We're Not Sorry" is meant to give the show a burst of energy before wrapping up. "I See A River" offered some delightful phony white girl funk, which Jennifer Laura Thompson delivered to a T.
If some other specific references occur to me, I'll update this post.
Begin at the beginning and go on till you come to the end: then stop.