Just looked through the DMG. Absolutely none of the potions, elixirs, or oils in there require concentration.
The potion of animal friendship says that it grants you the ability to cast a spell. That spell doesn't require concentration.
Of the other potions, all of the ones that name a spell (rather than describing a non-spell effect) specify that you gain the effect of the spell--not that you can cast it. The ones that give you the effects of spells that would require concentration even go out of the way to say that concentration isn't required.*
Gaining the effect of a spell is not a game element that requires concentration on your part, unless you are casting the spell in some way, or unless something specifically says it does.
*The sole exception is the potion of mind reading which grants you the effects of a concentration spell without spelling out that you don't have to concentrate. Given the way the rules work, that is likely a mistaken omission of the redundant "no concentration required" statement.
Only one potion? I thought there were more. Potions are quite powerful, much more than ever before. I hand them out sparingly due to their power level.