Drinking/administering a potion takes an action, as you cited from DMG at p.139. That's the general rule in effect, unless specifically contradicted.
You note that the DMG's description of the Potion of Healing doesn't mention that it requires an action to consume, and seem to be wondering if that omission is meant to signal something. But every other potion described in the DMG also omits any description of the mechanics of drinking potions: they just start with phrases like "when you drink..." or "for one hour after drinking...." So we can't read anything into the omission, or the general rule would apply in no cases. (DMG5e pp.187-8)
So why does the PHB call out the consume action in its Potion of Healing description?
The Potion of Healing is the only potion described in the PHB (p. 153); the PHB doesn't have a general "potions" section akin to the DMG's on p.139. So this is the natural place in the PHB to mention that interaction with the action economy1. Would it have been clearer to add to the sidebar on p.190? Maybe, but that's a sidebar full of free interactions, not examples of things that aren't-free-interactions-though-you-might-think-they're-just-the-same.
Sidebar: But I can quaff an entire flagon...
Yeah, this bugged my table too. We made in-world sense of it by saying that a magical potion is an... unusual enough mouth-experience to take a moment2 to consume. A flagon of ale... apparently the game designers assume our characters are more-inured to that experience than to the experience of drinking spells.
1 - it's also mentioned in the Herbalism Kit, but that's the least-natural place to talk about the potion's consumption.
2 - I think about trying to get my kids to take emoxicillin and Wish it only took six seconds =P
A Rogue cannot use Fast Hands on any magical potion
Magic items do not require the Use an Object action to be activated, but a separate action entirely
The following quote directly addresses how this feature is limited:
[...] If an item requires an action to activate, that action isn't a function of the Use an Object action, so a feature such as the rogue's Fast Hands can't be used to activate the item [...]
- Dungeon Master's Guide (page 141: Chapter 7 - Treasure: Magic Items: Activating an Item)
Mundane items require the Use an Object action to be used
Do note that the above quote applies to magic items and the general quote which applies to mundane items is found in the description of the Use An Object action:
[...] When an object requires your action for its use, you take the Use an Object action [...]
- Player's Handbook (page 193: Chapter 9 - Combat: Actions In Combat: Use An Object)
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:
Activating some magic items requires a user to do something in particular, such as holding the item and uttering a command word, reading the item if it is a scroll, or drinking it if it is a potion [...]
- Dungeon Master's Guide (page 141)
Some items are used up when they are activated. A potion or elixir must be swallowed, or an oil applied to the body. The writing vanishes from a scroll when it is read. Once used, a consumable item loses its magic and no longer functions [...]
- Dungeon Master's Guide (page 141)
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 be consumed and that it is a magic item, and so its activation is not a function of the Use An Object action. Thus, the Rogue's Fast Hands cannot 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 to draw the potion
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. These are, as far as I'm aware, effectively the same.
Best Answer
No.
Drinking a potion is explicitly an action.
You can draw the potion as a part of the action that you use to drink it, but you cannot drink it as a free action.
However, it takes an action and cannot be used with the Rogue's Fast Hands ability per the DMG(p141):
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. This is quite different from two instances of interact with object.