Ye Olde Inn

So I know that a Hero carrying a potion can use the potion on themselves at anytime during his turn, (it does not exhaust his action for that turn) but what about that Hero using potions on other (adjacent) Heroes during his turn?

For instance, sometimes my players will move adjacent to another player and "feed" them a potion, but is this against the vanilla rules? Do other house rules allow for this kind of potion use?

1. Heroes can drink Potions without expending an action to do so. More relevantly, they can do this at any time. The NA rulebook explicitly says "As a Hero, you may drink a potion at any time." on page 17. In particular, a Healing Potion may be used by a Hero who has just been reduced to 0 Body Points to save himself from death.
2. The rules do not have any procedure listed for when Heroes can trade equipment between themselves, but they do allow Heroes to share treasures found in Quests between themselves. The relevant text from the NA manual can be found towards the bottom of the left column on page 16. We can therefore extrapolate that the rules as written permit Heroes to share their equipment when the situation calls for it.

Adding points (1) and (2) together, we reach the following conclusion: There is no functional difference between "a Hero giving a Potion to another Hero for him to use it" and "a Hero using a Potion directly on another Hero".

The only caveat to the above statement, of course, is that a Hero cannot use Healing Potions on another Hero to save them from dying due to having reached 0 Body Points. This situation is explicitly prohibited by the NA rulebook at the top-right of page 23. For this to work, the Hero with the Potion would have to have given it to the dying Hero prior to him taking fatal damage.

Mohawk is correct. This then falls to what limits there are on passing equipment rather than on using the Potion.

This sort of thing is why people re-write teh rule books so much

As far as passing items, there is a rule in later editions allowing adjacent heroes to pass an item to another hero so long as neither hero is adjacent to a monster.

If on his turn, the Barbarian passes a potion of healing to the Elf with 1 body point, does the "drink this potion at any time" rule allow the Elf to drink the potion immediately, even though it is not currently his turn?

Any time means any time. Yes!
That's why you can also drink a potion as a reaction to dying.

*imagines the barbarian wearing a hat that hold beer-cans, but with Potions of Healing instead...

*imagines the dwarf carrying a bottle that says Potion of Healing, but it's brandy instead...

My ruling on potions:
A hero may pass potions (or other equipment) to another hero provided the heroes are adjacent.
A hero may administer a healing potion to a dying friend on his turn only. (Another ruling I use that makes this work is that you are not dead, dead until the dice pass to Zargon's next turn.) So if someone can get to you and they have a potion and open action they may administer a potion to save their friends life.
The hero saved in this fashion sacrifices their next turn. Dying is rather traumatic. Having been there myself and saved by extreme medical means...you have the wind knocked out of you. Now I am willing to imagine that healing magic will offset the majority of issues...but when you see the pearly gates and then snap back, that is jarring. Lose the next turn.

(This rule modification is a much needed player friendly modification as we always play it hard mode.)

I allow a healing potion or spell to be used to revive a dead hero like this,

If it administered before the heroes next turn, roll 4 combat dice and recover 1 BP per non skull. reducing the number of combat dice rolled per turn it takes to get the potion administered thus for my girls they have in theory a maximum 4 turns to help each other, but even then no guarantee the potion will work...

I guess it is sort of a bleed out mechanic...