Dental remains from the incinerated remains of Hitler were collected within a few days of his suicide, and positively identified by multiple dental personnel familiar with his teeth and bridgework. Although this was not admitted by the Soviets at the time, images of the remains would eventually be published in the 1960s in the West, where they were further corroborated against x-rays that were in the possession of the US government. In short, we have a very, very high level of confidence that pieces of his jaw and teeth survive in the possession of Russia.

Additionally, although not revealed until the 1990s, a skull fragment had been recovered in the area, with a bullet hole, a year later (1946) and tentatively identified as Hitler's also. The existence of this was first publicly acknowledged in the early '90s, and it would eventually be put on display by the Russian State Archives in 2000. However, testing done in 2009 showed the skull was not Hitler's, unless Hitler was actually a ~40 year old woman at the time of his death. When this was revealed, the State Archives claimed that Dr. Bellantoni, who had conducted the test in association with History Channel, had never actually visited the Archives or handled the skull. A note here, that I absolutely believe Dr. Bellantoni, and have every reason to believe the Archives are simply playing the ostrich to results they didn't like, even though in all honesty they ought to have expected it. Dr. Bellantoni also recovered very degraded DNA from the upholstery of a bloodstained couch from the Bunker which might have been Hitler's. This was determined to be male, but lacking comparisons (Hitler has some living relatives but they did not provide comparison samples) not much more could be said there.

Last year (and published earlier this year), the Archives finally again let a team access the skull fragment in a pretty naked attempt to push back against Dr. Bellantoni. They were given access to the skull and allowed to do morphological evaluations, after which they determined it was not possible to determine age or sex of the skull, so maybe it was his after all. But were not allowed to do their own DNA testing, which would have settled the whole thing once and for all. They were also the first outsiders given access to the dental remains, ever, but simply went by dental identification with known radiographies of Hitler to confirm the teeth were his, again not conducting DNA testing, although dental identification is considered reliable enough that it wasn't necessary anyways.

So in short, yes there are remains, and while testing of the dental remains, which we are pretty confident about, might yield results, the Russians are probably never going to let that happen, because they want to keep up the appearance of the legitimacy of the skull fragment too, and they can't very well let someone test the teeth without testing the skull.

However, as I have touched on some time back here, there is really no particular reason to give credence to claims of recent Jewish ancestry of Hitler.