We assume that Constructs have "guts of gears" or magically infused stone, clay, or other materials.
It is unclear if they even have a mouth or digestive systems.
We found other SE answers showing which creatures cannot use scrolls but found others suggesting wide leniency for many other types of magical items. But what about magical potions?
Can Constructs use magical potions?
For reference, the construct type is described in the basic rules identically to its description in the introduction to the Monster Manual:
Constructs are made, not born. Some are programmed by their creators to follow a simple set of instructions, while others are imbued with sentience and capable of independent thought. Golems are the iconic constructs. Many creatures native to the outer plane of Mechanus, such as modrons, are constructs shaped from the raw material of the plane by the will of more powerful creatures.
Nothing in the rules prevents constructs in general from drinking potions. Specific potions might state that constructs are unable to gain their benefits, but I haven't seen any such restrictions.
They are creatures, and can do anything that other creatures can do unless specified otherwise.
As always, DMs can houserule differently.
My answer applies to constructs in general. Particular creatures might have specific traits that might change the applicability of this answer to them.
Several monsters from the Monster Manual (animated objects, golems, helmed horrors, scarecrows, shield guardians) and Mordenkainen's Tome of Foes (cadaver collectors, clockworks, hellfire engines, maruts, steel predators, stone cursed) have the "Constructed Nature"/"Construct Nature" trait:
Constructed Nature. A [creature name] doesn’t require air, food, drink, or sleep.
However, as written, this trait simply states that such constructs don't require drink (e.g. water). It does not state that they are incapable of benefiting from magic items such as potions. As such, the trait does not prevent them drinking potions.
If the construct is able to swallow things (or do something that could be understood as swallowing).
The DMG page 141 (emphasis mine):
A potion or an elixir must be swallowed, or an oil applied to the body.
I didn't find anything about the importance of the digestive system. So it looks like the only prerequisite is the ability to swallow, not to digest. So at the end of the day it depends on the DM if some construct is able to swallow or not.
