I'm using a guide to learn how to setup my guitar and it recommends tightening the washers around the tuner spindles (the metal pieces the strings are wound around). Is it possible to over-tighten the washers and crack/dent/damage the wood around the spindles or will the washers become too difficult to tighten before any possible damage can occur?
Depends on how soft the wood of the headstock is, and how strong you are. I'm pretty sure you'd be okay, though, you'd really have to not be paying attention to overtighten those washers.
You don't tighten washers, you tighten nuts. But the answer to your question is yes, you can damage your guitar if you tighten too much, but that's the same for anything on the guitar. You should tighten it so they're snug, but if you're not sitting there cranking on the things then you'll be fine.
They aren't taking any major forces so they don't need to be cranked right down - just tighten them til they feel firm. If they feel like they are starting to protest at all stop.
Is there any benefit to tightening them besides the fact that they buzz if they're loose?
You just want the nut to be snug.