About a month has passed since the film premiered and a lot more information about the movie is out there now on Japanese wikis, blogs, yahoo answers, etc., so I thought I'd post some of the more detailed information. I'm posting it here instead of in the article as I have yet to see the movie, so I can't completely confirm all of this information.
So, basically, the term Tenseigan refers to two different things in the film, which has caused a lot of confusion.
The first is known as the Energy Vessel (エネルギー体, Enerugī-tai), and is the Ōtsutsuki clan's greatest treasure. This is the altar into which the Ōtsutsuki Clan had been sealing their eyes for a thousand years, and "Hamura's Tenseigan" (ハムラの転生眼, Hamura no Tenseigan) is its core -- which of course indicates that Hamura had awakened the Tenseigan at some point before sealing it in the altar. The Energy Vessel apparantly uses his Tenseigan to act as a giant senjutsu chakra engine that collects and amplifies natural energy, and the Ōtsutsuki clan used the Vessel as an energy source to power everything on the moon, including the countless autonomous puppets that served them.
Hamura had vowed to Hagoromo that he would wait a thousand years before judging mankind. The Branch House of the Ōtsutsuki clan interpreted this as a condemnation and believed that the Tenseigan was meant to be used as a weapon to destroy humanity for using chakra as a weapon -- though, in truth, Hamura's intent was probably for the clan to use it to restore the Earth in the event that mankind misused chakra and destroyed the world. The Branch House and Main House eventually destroyed each other in a civil war (apparantly sparked by Madara's theft of the Gedo Statue) during which the Energy Vessel was used as a weapon and apparantly became deactivated. Toneri (only a child at the time), heir to the Branch House, was the sole survivor, and reactivated the Energy Vessel at some point before carrying out his plan to fulfill Hamura's vow.
The Energy Vessel can apparantly only be touched by those with the "undiluted chakra of Hamura," in other words the Ōtsutsuki and Hyuga clans, and even Hinata, who was a direct descendant of Hamura, was only able to destroy it by sharing her chakra with Naruto for a combination attack.
The second is the actual Tenseigan dojutsu activated by Toneri, and Hamura before him. After Toneri transplants Hanabi's Byakugan, they react to his Ōtsutsuki chakra and transform into Tenseigan. Contrary to early reports and interpretations, this apparantly has nothing to do with the countless Byakugan sealed in the Energy Vessel altar and Toneri's Tenseigan isn't channeling their power (rather, the Energy Vessel housing Hamura's Tenseigan is, hence the confusion).
So, not unlike the Rinnegan, it's actually awakened by Hamura's two bloodlines uniting (Toneri's Ōtsutsuki chakra and Hanabi's pure Byakugan). Hanabi's Byakugan are especially pure in that, as an heiress to the Hyuga Clan's Main Branch, her bloodline is undiluted. The same is true for Hinata and Hiashi.
In addition to all of Toneri's dojutsu that have already been recorded elsewhere, it seems that he can probably use Senjutsu... which probably isn't surprising, given the appearance of his Tenseigan Chakra Mode and its similarities to Six Paths Sage Mode (Truthseeker Orbs, flight, etc.). At any rate, he animates a giant stone statue of Hamura to fight Kurama, and animating inorganic material certainly falls within the realm of Sage Mode (see: Kabuto). This is speculated to be the way their version of the Puppetmaster Technique works, as they're powered by the Tenseigan Energy Vessel which apparantly generates senjutsu chakra.
Anyway, hopefully the impending English theatrical release helps clear up things further... and hopefully we get some more concrete details on this stuff. The movie databook was unfortunately a bit light on details.
And, again, keep in mind that this information is all second hand, pulled from Japanese wikis and summaries, so I can't completely confirm its accuracy.