The Lubavitcher Rebbe has an unusual psak on what to do if you cross the international dateline during the Omer. He says to continue counting with a bracha according to your own personal perspective of the sunrise and sunset. According to him if you continuously travel around the globe, Shavuot can fall on any day of the year. Source
This of course opens a whole can of worms on what happens when Shavout falls on significant dates. Does Shavout override minor fast days? What about Shavuot on Yom Kippur or Tisha b’ av? What is the torah reading for when Shavuot falls on Succot? What happens if you go around the entire year, do you start two simultaneous omer counts?
But I think the biggest question is what happens if Shavuot falls on Christmas. There is a custom to stay up all night learning on Shavuot, but on the night of Christmas (aka Nittel-nacht) there is a custom to NOT learn that night. So what happens if they fall on the same night?
Of course this is only a problem for Chabad. Everyone else celebrates Shavuot on Shavuot regardless of Omer-related travel, and most other segments of Judaism view Nittel-nacht as a historical custom that is not binding today. For Chabad adherents: do you engage in Torah study on Christmas if it coincides with Shavuot?