Why does some guys lift their left/ right lip, when smiling or even not smiling?

That's how they are! Maybe there is a subtle difference in the muscles of one side vs. The other side in people who do this. Maybe they do this because one of their parents did it, and it's an expression they learned to imitate in childhood. There's no disadvantage that i can see; it worked pretty well for elvis.
Muscles. It is the result of their muscles being activated. You can do it voluntarily , but some develop muscle patterns and do it automatically. Ask them next time !