My Ape Escape Any% No IJ run stands at 1:28:19, but I recently beat it twice; once with a 1:22:17 and most recently with a 1:17:55. My first run got verified within one day, but my first update was still awaiting verification by the time I had improved it; I subsequently deleted the request and added a new one. My question is this: Are run verifications taken seriously if they merely improve the time of a user instead of beating someone else's time or getting World Record?

My main concern is not getting my time up, but rather not annoying the moderators with my requests

Every PB is equally important. Anybody not taking them seriously in verification is a problem. You also don’t have to delete runs, submit them as you PB and once they’re all verified they’ll be in the right order.

Getting runs verified can take up to a couple weeks depending on how busy the mods are. If your runs sit for a very long time, you can always politely message a mod and ask them about it. Just don’t do that frequently or after only a couple days as that’s an easy way to annoy someone.

PBs are very important last place, first place, in the middle all of them matter equally ^_^

Also worth mentioning that although you're not changing placement now, in the future someone could submit their own run, and your PB improvement could be the difference between you being ahead of or behind them.