Tennō die, and are buried. Their tombs are important in Shinto practice, and are under the control of the Imperial Household Agency.
In myth, the first Tennō, Jinmu, became Tennō in 660 BC, and the current Tennō is the 126th. All of these Tennō (apart from the 125th and 126th, who are not yet dead) have official tombs, and most of these tombs are fairly large mausolea, or tumulus earth mound graves. In some cases, there is no doubt that the tomb is genuine. For example, Shōwa Tennō died in 1989, and is buried in a tomb called Musashino no Misasagi, to the northwest of Tokyo. Reliable records do go back several centuries, because the tombs have always been important, but there are some dubious cases from periods of civil war, and if we go back far enough, there are even greater reasons for doubt.
For example, the largest tumulus tomb in Japan, the Daisenryō Kofun, is supposed to be the tomb of Nintoku Tennō, the son of Ōjin Tennō. The tumulus (“kofun” is the Japanese for “tumulus grave”) is 486 metres long, and is dated to the late 4th century. This is about the right date for Nintoku Tennō, and it seems likely that the tomb is for someone very important, as it is the largest such tomb in Japan, and possibly the world. (For comparison, the Great Pyramid of Giza is only about 230 metres on a side, although it is higher.) However, there is real doubt over whether Nintoku Tennō actually existed, and if he didn’t, this is probably not his tomb. (Incidentally, Ōjin Tennō is commonly identified with Hachiman Ōkami, possibly the most popular kami in Japan. His tomb is not far from Nintoku Tennō’s, and not much smaller. Quite a few kami have tombs, and in some other cases the identification is historically secure.)
The issue becomes worse with Unëbiyama no Ushitora no Sumi no Misasagi, which is said to be the tomb of Jinmu Tennō, the first Tennō. Historically, he did not exist, and the tomb is much later than 660 BC. Nevertheless, it is officially recognised as his tomb.
Most (maybe all) Imperial tombs have torii in front of them, and space for people to pay their respects, just as they do at a jinja. Indeed, the idea is that you are paying respect to a kami — the Tennō — so the actions are naturally the same. (Technically, then, Ōjin Tennō’s tomb can be considered to be a Hachiman jinja, but it really is not thought of in that way.) The Imperial Household Agency, which is part of the government, has a section that looks after the tombs, and the Tennō’s personal ritual staff perform rituals there on death anniversaries. At least on important anniversaries, such as centenaries, members of the Imperial family may attend these rituals, and Jinja Shinpō always reports the major ones.
When members of the Imperial family do important things, they normally visit Jingū, to report it to Amaterasu Ōmikami, and Jinmu Tennō’s tomb to report it to him. They may also visit the tomb of Shōwa Tennō. The imperial retirement and accession were reported to these locations, and to the three Tennō before Shōwa Tennō.
Thus, although it rarely makes the news outside, or even inside, Japan, the veneration of Imperial ancestors at their tombs is still an important part of the practice of the Imperial family, and an aspect of Shinto that the Shinto establishment regards as important.
In October I had the pleasure of being able to visit Koyasan and took a tour of Okunoin Cemetery. When we were approaching the Kobo Daishi Mausoleum our guide pointed to a segregated area with tomb mounds (to the left/west as you approach the mausoleum), indicating that those were imperial tombs. Do you know if these fall under the topic and practice you describe here, or are they more secondary tombs tied to the bodhisattva incarnation that is supposed to happen in about a billion years?
I’m afraid I don’t know. It isn’t out of the question that some Tennō might have been buried there, as most Imperial funerals were Buddhist, but I’ve not, in my quick attempts to research it, been able to confirm whether it actually happened. There is also a real possibility that the temples have an ancient tradition that Imperial burials happened there, but that this tradition was over-ruled by the Meiji government in its attempt to purge Buddhist elements from Shinto, and claim the Imperial family exclusively for Shinto. As this is not one of my primary areas of interest, I don’t know. Sorry.