Rothenburg is not a nightlife town. The day-trippers clear out by early evening, many of the shops and tourist facilities close, and the old town becomes quiet and, in the best possible way, its own self again. What remains is a small number of wine bars and cafes oriented toward the overnight guests rather than the coach crowds.
The thing to drink is Franconian wine — pale, dry whites from Silvaner or Müller-Thurgau grapes, served in the squat round Bocksbeutel bottle that is the symbol of the Franconia wine region. Local beer exists, but wine is the right call here. A few of the old inns and wine bars around the Marktplatz and on the quieter streets toward the walls stay open after 9pm and are worth finding.