Honestly not that many, for the reason that Indiana Jones does not really engage in actual archaeology all that much. I will expand on that, but just to go through the list, despite "it belongs in a museum" being his catchphrase the only time he actually acquires an artefact is the Golden Idol in Raiders (granted I haven't seen Temple of Doom in ages). That, certainly, was against standards of the time and today, while I cannot speak to the country where the temple was as it is not specified, antiquities laws were common across the world by the 1930s--for example the beginnings of regulation of that in Egypt date back to 1835. Belloq's removing of the Ark from Egypt would certainly be illegal and professionally censured today, but it is possible he did so with the approval of the Egyptian government. The rest of the movies are much the same, he does plenty of illegal things like breaking through the floor of a Venetian Church, but that is not specifically an archaeological no no, that is just vandalism.

The broader issue is that Indiana Jones is not doing any archaeology, really. Archaeology is the study of the past through material remains. This is often done through excavation, in which soil etc is literally removed in order to expose an ancient surface, but it also overlaps with architectural history and involves techniques like field surveys where little if any digging is done. The actual objects found are not really the point, they can provide a useful window into understanding past societies but that is what they are--windows. The point is to understand the past society, not to find cool objects (although, of course, cool objects are cool). Indiana Jones, on the other hand, is looking for specific named and documented artefacts, which is a fundamentally different exercise.

While more could probably be said on the topic (for example the conflicting interpretation from u/tiako here), u/CommodoreCoCo offers a very thorough analysis of Jones' archeological practices here. A large part of the disagreement here seems to be what context to take Jones' work in, as CommodereCoCo identifies the work we see Jones doing as some mix of surface and salvage archeology while Tiako identiefies it as all the research/work Jones was planning to do. It would be somewhat interesting to see a discussion on this between the two of them, as I suspect that the disagreement is not just based on review of the films but also the contemporary norms of their given specialties (with the surface of Italy being a bit more worked over than that of The Andes apart from Greater Bagota).

I would suggest visiting with the same question /r/askanthropology.

So, in Indy’s defense many of the artifacts he looted were about to be looted by someone else… I would rather he stole them then the Nazis.

It’s true he doesn’t have any screen time devoted to actual archaeological work, although the 5th movies hints that he was doing that when he was captured at the beginning with the clay artifacts they say he was “digging for in the desert.” If we assume he wasn’t just looting them, then it was part of his work as an archaeologist. He could have had a mostly normal career as a professor of archaeology and these were the only events that were interesting enough to be made into movies.

Also in the first movie I think in his opening lecture maybe he was describing a site he worked at? And and he spoke about the destruction of looting. Indy strongly believes in the importance of preserving the artifacts for public benefit, if we believe his “that needs to be in a museum” so despite the destruction that takes place during the adventures, it seems to be the main priority to simply safeguard the most valuable artifacts in the face of opposition.