Future Lister is an alternate version of Lister from a future that was prevented. This version, like all of the Future Dwarfers, was immoral and corrupt due to years of abusing the time drive.
Future Lister's life was identical to Dave Lister's up to when the Starbug crew found the time drive. Either they never met their future selves, or they allowed them to copy designs from the time drive. However, it is known that the crew continued to use the time drive and abandoned their search for red dwarf. The reasons for not continuing the search is unknown, as they could have simply gone back in time to intercept red dwarf.
After presumably finding a device to travel quickly through space, the four began exploring history for their own reasons. Apparently, they never tried to help others or prevent the horrors humanity has faced, potentially refusing to get involved with anything. According to Future Rimmer, the crew had all lost all their redeeming qualities and become immoral selfish people who has befriended all the worst people in history.
However, Lister's body was presumably damaged from an accident, though the possibility of it being a real accident is debatable. To save him, the crew apparently placed his brain in a jar and secured a device that enables him to speak.
Eventually, their time drive malfunctioned, limiting the crew with only travelling back in time. The future crew attempted to get aid from their past selves, but their rude and snobby personalities along with their friendships with people like the Hitlers forced the younger crew to abandon them. Unwilling to go back to how their life was before, stranded in deep space lacking full control over time, the future crew attack the younger versions, wanting to force them to give them access to the time drive, despite knowing that the deaths of their younger selves will kill them also. Unwilling to surrender and become them, the younger crew fight back, resulting in all but Rimmer dying, who proceeds to destroy the time drive. The resulting paradox seemingly erases the future crew, Lister included, from existence, with only the memory of them existing and their fight with their younger selves shown through the dimensional anomalies caused by the paradox.