... If one tube in a push pull pair has no input signal (e.g. by grounding its control grid), does this cause problems in the output transformer?
No problems as long as you don't disturb bias. You can directly ground the grid of a cathode biased amp, but not so with a fixed biased amp. For those you'll have to kill the input signal some other way.
Envision a Volume control ---> Coupling cap ---> Bias supply + Grid
Simply "turn down the volume" on one side of the push-pull output. That side still idles, but has no AC-input.
... the output transformer only becomes mismatched when both tubes are pulling different amounts of DC? ... one has the additional AC riding on it, which doesn't matter because the OT only cares about ...?
Be careful with the word "mismatched." Mismatched to what?
At lower power-output levels, each side of the push-pull output sees 1/2 of the total "plate-to-plate impedance" of the primary. At higher output levels in Class AB, one side is cut off and the remaining side sees 1/4 of the total plate-to-plate impedance. That allows the side that is on to flow more current for the same AC-voltage-drop (which is most of how Class AB delivers higher power output that Class A).
Turning off the drive to one side of the push-pull power section but retaining the DC idle current effectively never "shuts off" that side of the transformer primary. The driven side always sees the "1/2 of the total plate-to-plate primary impedance."
Now whether that load is "optimal for max power output" or not probably doesn't matter, because the user wanted less power anyway. And the tube doesn't really care what its loading is, it is simply attempting to pull current through the primary in response to an input signal. Therefore "mismatching" is an arbitrary and/or academic notion, and best forgotten (it's common due to the way people first learn about transformers, but the concept is over-emphasized past a point where it stops being useful).