In the manga, at some point in time Future Beerus participated in the All Universe Hide and Seek Tournament held by Future Zeno. But when he hid and fell asleep for fifty years during the tournament, it had to be cancelled. Future Zeno, who was enjoying the event, became furious and had to be calmed down by the gods from the 11 universes. Since the other Gods of Destruction have felt disdain towards Future Beerus.
Dragon Ball Super
"Future" Trunks Saga
Main article: "Future" Trunks Saga Future Beerus' life was the same as his counterpart until Age 766 when Future Goku, who had the potential to become a Super Saiyan God, had died from a Heart Virus, and the Androids killing Future Vegeta and Future Gohan, 13 years later, thus the prophecy of the Super Saiyan God did not arise in his head. Due to Goku's death, he never visited Earth and remained asleep indefinitely on his planet.
As a result of Future Shin's death at the hands of Goku Black (Future Babidi and Future Dabura in the manga), Future Beerus was also killed, presumably while still asleep. His soul was destroyed when Future Zeno erased all Universes from the timeline.
Alternate Future Beerus
Prior to his death, he was visited by Present Whis to be warned of Future Zamasu's Zero Mortal Plan so he could deal with him. This prevents the plan from being carried out in a new timeline for Future Trunks and Future Mai free of Zamasu's reign.
In the manga, Future Trunks and Future Mai travel back to a time before Dabura killed Future Shin in order to prevent their deaths. This action was approved by the main timeline Beerus since it meant saving him from certain doom.