WASHINGTON—Presidential candidate Donald Trump’s call to ban Muslim immigrants is straining his already shaky relationship with the Republican establishment in his home state of New York.

Joe Lhota, the 2013 GOP mayoral candidate, is calling on the Manhattan Republican Party to kick him out, relying on an obscure election law that allows a local chair to cast out a voter “not in sympathy with the principles of such party.” Republican consultant E. O’Brien Murray is considering filing a complaint with the party.

The...