@MrBarbarroja,
The honest answer is
it depends on how the headphone is wired as most headphones are actually
balanced contrary to popular belief otherwise you wouldn't get separated left & right channel sound
(stereo) & the whole thing will be mono in any case, not stereo from that single cable entry.
Generally, when a headphone has dual entry cables, all this usually means it the grounds in the wire as separated so technically one should get a
cleaner signal though once again, this depends on how good the manufacturer is at wiring as there are a lot of single cable headphones which technically work better than dual entry.
Also if the headphone connector plug into the headphone is long enough & wired for balanced, eg 3.5 Balanced ending in a Balanced termination, eg XLR, 3.5 Balanced, etc, there should be no real issue why it can't work though as said it will depend on how both headphone & cable with said balanced plug is wired.
So in answer to your question, no, there should be no damage to either headphone or amp as if said cable &/or headphone isn't wired for balanced, only one side will work (usually the right side though sometimes it is left side) if it can be plugged into a socket, provided it can be plugged in the first place.
However, saying that, on the safe side, it is important to check what cables & what headphones can take balanced or not, just in case.
Also, always make sure to have the source volume at zero before unplugging & plugged in headphones to avoid feedback issues, some manufacturers will also state not to leave headphones plugged in to said amp when off as well to avoid electrical discharge when powering on said amp.
Hope this makes sense.
Hope you have a great day !