What would you call someone who enters old places, like ruins and sells the stuff he finds?

Not as a profession, but usually pejorative, that would be a scavenger. Dictionary.com has:

a person who searches through and collects items from discarded material.

Probably a term that is worth citing, giving the amount of ancient ruins present in Italy, is from the Roman dialect: tombarolo. The term is used also in English:

Tombarolo is an Italian term (plural ‘tombaroli’), derived from the Italian word ‘tomba’, meaning tomb or grave.

It refers primarily to ‘tomb-robbers’ operating in Italy (Mackenzie and Green 2009: 3), although the term ‘clandestini’ (meaning ‘illegals’ or ‘clandestines’ in Italian) has also been used to describe clandestine diggers (Renfrew 2000: 17), and a tombarolo might not necessarily limit their activities only to tombs and graves; for example sometimes removing material from villas and other sites (Watson and Todeschini 2006: 151).

(traffickingculture.org)

Tombarolo:

In Italian contexts: a person who steals archaeological artefacts and antiquities from tombs or graves for sale to collectors; a grave robber.

(Lexico.com)

Perhaps Treasure Hunter?

This person is not a scavenger, "dumpster-diver," garbage-picker, or the like.

Often they're looking for something with real historical value, and may keep or sell it.

Depending on where they're entering, and what they're taking, it could be criminal! For example, taking pottery from an ancient burial ground carries some hefty penalties in the United States. https://en.wikipedia.org/wiki/Treasure_hunting

If you're building a character, they have a certain "character," if you know what I mean. They're sharp-eyed and snoopy. They're always looking to get their hands on something nice for free or on the cheap.

Your question reminds me of an acquaintance who was always going to estate sales, garage sales, sizing up piles of garbage, etc. He wasn't a scavenger; he had a good eye for stuff. One day he found a Beatles lunch box in the garbage; he turned around and sold it for $300!