The rouge was probably going to die anyway,

I think we ended up killing two rogues the last game I played. Well, we killed one and sort of made no effort to save the other.

Typically they heal from necrotic energy instead of positive like living things. So a healing potion for a lich is effectively a vial of pain juice for something living.

Liches are undead, so that potion would probably rejuvenate necrotic flesh or it's a strong poison to create more necrotic flesh. Both are not something supportive of living like a health potion would be.

Technically in 5e, there's no reason they can't. Undead can benefit from potions just the same as anyone else. And given they have to make a potion to transform into a lich in the first place, I'd say they probably are skilled alchemists, so it'd not be out of line to say they make some.

Previous editions probably had other rules against it though, and it wouldn't work for pathfinder I believe, as most healing potions have a positive energy trait (which harms undead rather than heals).

True. However, a lich's "healing potion" would likely be a potion of Cause Critical Wounds, not healing. Whether or not it would be visibly similar to the Heal version would be up to the DM, I suppose. I don't remember reading any specifics of that in 1st, 2nd, or 3rd.