My only experiences with HF transmitters and garage door openers had to do with the transmitter making the garage door open when we did NOT want it to do so..... lol
also, there was a lighted china display cabinet, whose lights would flash in time to whatever music was being broadcast on the HF-AM transmitter-- I solved that problem by unplugging it. lol
As far as damage caused by an HF transmitter to garage door opener frequencies, it is not likely-- but proximity to transmitter antennas, and the power being used, can play into it.... it is not impossible, either....
However, Ultra-- in this case I don't believe that there would be "partial damage" which would only affect the "range" of the remotes--
I believe they would either work, or not work, entirely--- like one band of a radio shack SW receiver that I accidentally burned up when opening the transmitter up on an antenna in close proximity to the receiving antenna-- in that case, I could actually SMELL the burning in the shack--
Anyway, I think the suggestions of 'interference' that others have made here are more likely correct...