Some times we do, but not always willingly.

In recently years there was considerable outrage in the Chinese media about artefacts being auctioned which were looted from the Summer Palace when the British and French marched on Beijing in 1860. The palace is still in ruins from it's coordinated destruction by British forces, so having people further profit off of this was more than a little upsetting. For a time there, quite a number of items were getting press when they came up for auction. In a number of cases wealthy Chinese nationals were showing up to bid on the items so that they could get them back to China. There were a number of scandals though, with at least one case where the winning bidder simply refused to pay (and also therefore never received the item in question)

In April this year, it happened a little more peacefully. There had been a pretty major smuggling ring bringing artefacts out of India and, once it was discovered, some of the museums which had unknowingly (we'll assume) purchased these items have since begun to return them to India. Not all though. This is still an ongoing issue

Sometimes it goes the other way, though. In 1980 a Chinese house (the whole house) from the 1700s was sent to Salem Massachusetts with the approval of the Chinese governmental actors.

There are issues though. Some people will argue that the artefacts are not safe in their original location. With groups like ISIS destroying things, you'll have a tough time convincing any museum to ship items back to Syria or Iraq. Groups like ISIS and the Taliban is not alone, though. You'll find similar justifications given around the world for why artefacts should not be returned.

Then there's also the questions of should they be, even if there were world peace tomorrow, since they have some educational significance by being far from their origin. If my childhood museum only had things from my hometown, it wouldn't really be that educational. But instead you get to see Ming furniture and mummies and Navajo art. One could certainly argue for there being value in that kind of exposure.

The question of returning items isn't really as straightforward as you might think.

Are you talking about returning them to their specific original context, or to the country in which they were found?

In the United States, certain artifacts are returned to the descendants of their owners, due to the requirements of the Native American Graves Protection and Repatriation Act of 1990 (NAGPRA). The kinds of things that are returned to Native Americans include anything associated with humans burials that are affiliated with Native American people (including the remains themselves, and any accompanying artifacts), "sacred objects," and "objects of cultural patrimony." The latter two categories can be contentious, to some degree (they depend a great deal on the exact definition of what is sacred and what is patrimony), but in my experience the law is followed conservatively by the vast majority of archaeologists in the country. Certainly though, there are still attempts on the part of interested parties (archaeologists, museums, Native American tribes) to test the boundaries of the law (most prominently the "Kennewick Man" case).

There are also laws and treaties governing the importation of artifacts from other countries, the movement of artifacts across state lines, and the sale of artifacts. in many cases, these laws also result in the repatriation of the artifacts in question. I'm less familiar with these laws, though, so I won't attempt to get into details.

Source: National NAGPRA: Laws, Regulations, and Guidance. Electronic document: http://www.nps.gov/nagpra/MANDATES/INDEX.HTM