Nighteye has a quirk which allows him to foresee the future and when he does (like in the case of All Might) he tries to do something to prevent it from happening when it's something bad. Why couldnt he foresee his own future or did he see it and did nothing?
First, Nighteye's Quirk requires him to look into the eyes of the person whose future he wants to see. It's not stated whether non-direct eye contact works, but assuming it doesn't then he can't see his own future just by looking into a mirror.
Second, he didn't think the future could be changed, only reacted to. When we first see Nighteye demonstrate his Quirk, he uses it on Deku so that he has perfect knowledge of his future movements. When they fight, he doesn't try to get in Deku's way or otherwise try to change those movements, he just uses that knowledge to avoid being where Deku is. Similarly, he himself said that he regretted seeing All Might's future because he's never been able to change what he sees - so presumably, if he saw his own death he would assume that it was a fixed possibility (and, in fact, in the fight with Overhaul it seems like he has used his ability to see his own death from someone else's perspective, and assumes that it's a foregone conclusion). It's only when Deku beats Overhaul and doesn't die in the process, as Nighteye saw, that he realises that the future can be changed.