How does setting a dimension to expire after a hit vs a visit impact reporting, specifically traffic metrics like visits, pageviews, instances, and occurrences?

I assume you're asking about evars specifically, because all props expire after the hit where they fire. But with evars you can change the expiry to be hit, visit, purchase, 30 days, etc. 

The difference for the expiry is how long the value is going to be remembered for. For example, say I'm capturing search term in an evar with a purchase expiry. I have one hit where the search term fires, then a product view, cart add, checkout, and place an order. Because the search term expires on the purchase, it will get credit for the search, product view, cart add, checkout, and order. If I have it set up as a hit expiry, it would get credit for the search and that's it, because then it would be forgotten once I move onto the next hit. 

When looking at something like page views (or visits) with evars, it will count all of the times it fired and when it persisted. So with our purchase expiry search term, it would get multiple page views associated with it (initial search, product view, cart view), but a hit level would only get the initial search.

As for what it makes sense to set evars as - that depends on how you want to report on them. Do you want to be able to attribute orders to different search terms or campaign IDs? Then you should set them to expire on purchase. Do you only want to see the initial number of searches conducted and don't care what they did after? Then you would use a hit. 

Best thing to do is take a step back, figure out what question you're trying to answer, and then think about what data you need to answer it. That will help you figure out how you should set up your evars.