(This is such a difficult question to begin answering in any satisfactory manner, though I think can be addressed for pedagogical purposes in line with the aims of this SE.)
Before we address the question, I think it is prudent to think about the difference between stimulation and activity. A neuron can be self-activating without external stimulation, or it can be highly stimulated by sensory input and yet have low or inhibited activity. You do not specify, so I think the discussion must necessarily be very incomplete on this topic.
Your question is also naturally a little unclear and difficult to answer methodically, especially because 'damage' in this case is a very subjective term. As with all biological systems, there are self-protective mechanisms and reflexes and feedback circuits in place to prevent 'you' or your brain from overdoing anything. A dysregulation in neuron activation, or activity, would probably be a state of illness, such as having a seizure. I don't think it is possible for a typical person to expose themselves to sensory extremes without consequently feeling tired or bored as a result (e.g. looking into a strobe light or listening to loud music for inordinate amounts of time), which may be the result of an internal homeostasis, just like blood pH or blood sugar level, though regulated through physiologically-dependent behavior (sleep, concentration levels, apathy, etc.)
More strictly and scientifically speaking, all activity wears all tissues, including the brain. This would make 'brain activity' a process contributing to aging. However, once again, one can interpret anything as 'damaging'; breathing is damaging (oxidation! the cellular definition of wear-and-tear), but it also sustains you. Calories sustain you, but caloric restriction extends lifespan in virtually all animal models.
A recent publication in Nature (Oct 2019) suggests that the global inhibition or decrease of all neuron firing seems to be related to increased longevity which may suggest the opposite case to your question: lower neural activity may lead to extended lifespan. This suggests, though does not demonstrate, that over-activation may contribute more 'damage' and accelerated aging, compared to lower levels of activation. This conclusion is also based on evidence associated with a global - whole-central nervous system - effect, and does not indicate that one should under-stimulate parts of their nervous system to achieve the same result. Also, keep in mind that stimulation and activity are entirely separate things, so such a conclusion does not answer your question.