I'm blinded and have a health potion in hand that I want to administer to an ally in an adjacent square.
Do I need to succeed at a DC 11 flat check due to the ally being hidden from me? If yes, would I lose the potion if I fail the check?
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..must succeed at a DC 11 flat check when targeting you with an attack, spell, or other effect or it fails to affect you. Area effects aren't subject to this flat check.
Effect is a defined term... sort of. Effects are literally everything that happens.
Anything you do in the game has an effect...
So attempting to use a potion on an ally you can't see is subject to Hidden's flat check.
By-the-book, you must have used the potion in order to make the Flat check.
Unless stated otherwise, it's destroyed after activation.
But this is an oddly specific edge case. It might not make much sense to pour an entire potion out in an attempt to find an ally's mouth. This is a prime example of where GM's can apply the core principle of Creating a Narrative
the players take turns describing what their characters attempt to do, while the GM determines the outcome
If it doesn't make sense, GM's can and should rule that what makes more sense happens (particularly, I believe, in cases where it helps the PC's succeed). In this case, it could go either way. Perhaps you poured it into what you thought was a mouth, but was actually a fold in their cloak. Or perhaps you simply failed to find their mouth and had to give up because combat kept moving. If you're out of combat trying to give a potion to an invisible ally, that changes the narrative as well.