Except being stabbed by Dragonglass/Valyrian Steel which can kill the White Walkers, in addition to other things they can be killed by that we might not know at the moment, are White Walkers in Game of Thrones immortals, i.e. if we don't kill them by something could they stay alive forever?
This can't be answered based on currently available information, either book, show, or interview.
White Walkers can be killed by the mentioned weapons, so they are not completely immortal. And nothing has been stated about biological immortality, if they can live forever if they are not injured or otherwise externally interfered with.