The border crossing between Tibet and Nepal has re-opened as of September 2017.
Detailed post by The Land of Snows.
It is currently impossible to cross the border.
(unless you hold a Chinese or Nepali passport.)
Recent info from The Land of Snows (which I have praised in an unrelated answer) as of 27 July 2016 (updated 23 Jan 2017):
Since the April 2015 Nepal Earthquake, the Tibet Nepal border has been closed to foreign travelers. [...]
At this time, it is impossible to say when this border will reopen. I expect the border will remain closed until at least summer 2017....maybe longer.
In a more recent comment as of 27 May 2017, Lobsang writes
The border is only open to Chinese and Nepali passport holders and is CLOSED to all other nationalities. The border remains closed and no one knows when it will reopen.
And this is still all that can be said. You may have gotten hopeful by this May 7, 2016 report linked in a comment by @JanDoggen above - however that precedes a massive rockslide just past the Chinese border buildings following heavy rains in July 2016 at Kyirong.
You should give a good look to the above linked post and especially the sections "When will the Tibet-Nepal Border Reopen?" and "What Areas Are Closed?"
The author of the website/blog promises to update that article once there are changes and post updates on Facebook as well.
The old border in Zhangmu seems to be a ghost town by now and apparently there are no plans to re-open the border. While residents claim the street condition on the Nepal side is as before the earthquake, there is still the occasional (fatal) boulder that comes down, as reported in this 8 months old article.