Quick thoughts:

• people will try to use game theory to solve this
• but the usual simple game-theory models are not realistic, because people sometimes care about not hurting other people (which means that the "pain" in my "opponent"'s outcome matrix also translates to a small amount of my "pain") and sometimes they also think long-term so they may give up an unimportant "fight" now in order to increase a chance of better cooperation in the future
• but doing this properly requires me having a good model of my "opponent"'s mind, so I can see how much "pain" various outcomes give them
• but I don't have a direct insight into my "opponent"'s mind, and this gives them an obvious incentive to lie and exaggerate their "pain" (and even if I could read their minds, they could self-modify to actually feel more "pain" if they knew it would make me give up)
• maybe there is a higher level of game theory that can deal with this situation
• but I don't know it.

In Yvain's post (linked here by gjm), Yvain says:

Although people pretending to be offended for personal gain is a real problem, it is less common in reality than it is in people's imaginations. If a person appears to suffer from an action of yours which you find completely innocuous, you should consider the possibility that eir mind is different from yours before rejecting eir suffering as feigned.

Uhm, it depends. My guess is that such people are rare as a fraction of population, but if they are skilled at exploiting other people's empathy, they can lie about their internal "pain" pretty often. So while the probability of "person X randomly chosen from population would do this" is very small, the probability of "a person who replied online on your post, acting offended and citing political arguments and calling in their numerous supporters, would do this" could actually be pretty high. (Prior probability, posterior probability, selection bias.) So I would probably use the way I have received a complaint as an evidence.

Situation A: I play a music I enjoy, and my neighbor says: "Excuse me, my ears hurt, could you please turn down the volume?" I would turn down the volume, and if it is too quite for me to enjoy, they I would simply turn the music off, or consider using headphones.

Situation B: There is an active political or religious movement X with typical modus operandi of finding something they complain about. My neighbor is a very active member of X. This month, their topic is "make your neighbors turn down the music, because our great prophet said music is sinful". I play a music I enjoy, and my neighbor says: "Excuse me, my ears hurt from your sinful music, you should be ashamed of yourself, and you will burn in hell. Could you turn down the volume?" I would ignore them, or offer a trade (something like "I am doing you a big favor here, and I expect some favor in return in the future"), depending on my mood and my estimate of their probability of returning the favor (the more righteous they are, the less likely).