Magic items do not require the Use an Object action to be activated, but a separate action entirely
Mundane items require the Use an Object action to be used
One specific example given, that is similar to drinking a potion, is that one can drink all the ale in a flagon using their object interaction.
Potions are magic and explicitly need to be "activated"
The question then is whether a potion requires an action to "activate", luckily the nearby sections on "Activating a Magic Item" and "Consumables" state the following:
Both of these quotes establish that drinking/consuming a potion counts as activating it, and we know that a potion of healing requires an action to consumebe consumed and that it is a magic item, and so its activation is not a function of the Use An Object action. We can conclude then thatThus, the Rogue's Fast Hands feature cannotcannot be used to activate (drink/consume) a magic potion.
Though drinking a flagon of ale and drinking a potion seem incredibly similar there is a key difference: one is magical. How this difference looks like narratively, or why magical things are harder to drink is up to your GM.
That, said, Fast Hands can still be used the same as on any other turn
We already have the following question and the highest-scoring answer there says:
[...] Thus you don't have 2 instances of interact with object going on, you have 1 interact with object (drawing the potion), and a second action that is specific to the magic item in question. [...]
What this means for our Thief is that they could replace the object interaction used to draw the potion with their bonus action from Fast Hands. This would let them still have an available free object interaction to use, likely during their movement.
Of course, they could also have replaced the object interaction during their movement with the bonus action from Fast Hands and used their free object interaction to draw the potion.
So while Fast Hands doesn't specifically help with the potion, drinking a potion also doesn't prevent Fast Hands from working as it normally would.