Unseen Servant Interacting With An Object
Per page 284 of the PHB, 5e, second paragraph of the spell description.
"Once on each of your tuns as a bonus action, you can mentally command
the servant to move up to 15 feet and interact with an object.
The servant can perform simple tasks that a human servant could do,
such as fetching things, cleaning, mending, folding clothes, lighting
fires, serving food, and pouring wine. Once you give the command,
the servant performs the task to the best of its ability until it
completes the task, then waits for your next command.
It is reasonable to rule "Yes, the Unseen servant can pour healing potion into the mouth of another party member" who is not actively engaged in melee combat(moving about, swinging weapons, blocking, dodging) since that is:
- No more complicated than pouring wine into a glass
- Within the complexity range of tasks described, and even simpler than mending clothes.
- The spell description looks to limit this to one bonus action per
potion delivered. Issuing "standing orders" and assuming that the
Unseen Servant will do it each round without further instruction
looks to be beyond the scope of this spell.
- With a Strength of 2, the Unseen Servant's carrying capacity is 30 pounds. (p 176 PHB) If a pint is a pound (good rule of thumb based on the weight of water) ... add X for what each potion container weighs to arrive at a case, a twelve pack, or a six pack of potions. Work with your DM.
- If using variant encumbrance, it can carry 10 pounds with no movement penalty, and up to 20 pounds with speed reduced by 10 to, and up to 30 pounds with speed -20. (p. 176 PHB). With a speed of 15, the max that it can carry will be a bit less than the 30 pound max. (Speed = 15 is based on 15 foot move in one turn from spell description). Again, work with your DM to determine how many bottles or flasks of healing potion is a workable load.
Your DM may rule (reasonably) that the Servant cannot pour the potion into the mouth of the party member if the member is actively engaged in melee. That "moving wine glass" (the PC's mouth) can be missed, spillage, etc.
Work with your DM. You should be able to arrive at "go" and "no go" situations where this service could be provided. I think it's a great idea personally, and would be open to that service at my table.
See also this discussion of the spell