Can conquerors Haki give someone brain damage? I always wondered does conquerors Haki have any long term effects like brain damage?
If you are knocked out by the Conqueror Haki you are no longer susceptible to other non-physical influences from a biological point of view. You can only be receptive once you are conscious.

Unfortunately, the anime / manga does not support my thesis, as in both the Conqueror Haki was not used several times in a row.

It can not be said at this point that Luffy's battles in which he competes against strong opponents and the impact of the release of the Conqueror Haki made a similar comparison with the deliberate use and switch off by the Conqueror Haki are comparable.

Conclusion: I do not think that this could cause damage.

Anyway, in the anime sometimes you see people slowly getting up but moving very slowly from the effect of the Conqueror Haki. I would say that if the character is strong enough you could at least speak of a "long-term effect" if the disabled people take a long time to get fit again. Maybe it would be possible to deprive people of their full physical abilities for days.