For the sake of scope, we'll assume the warrior already have the potion in their hand.
Also for the sake of scope, we'll assume there's no magic to help a person drink faster. It all relies on technique.
In combat, time is everything. If two warriors fighting each other drink a healing potion, but one takes 1 second to chug down the liquid while the other takes 3, the slower one will probably not survive, all else being equal.
Potions are stored in ceramic or glass vials with a cork stopper, and are standardized to 100ml (3.38oz) of volume. Vials are on average 11cm (4 1/3in) tall (including the cork) and 44mm (1 3/4in) thick. There's 1 or 2 mm of difference in each direction but every potion has 100ml of liquid.
Some movements are obviously a waste of time. For example, spitting the cork before drinking. Some props may help move the liquid faster, like putting a straw (of the hollow drinking kind, not an actual piece of straw) inside the bottle but you waste time putting the straw inside. These are okay but anything that delays the consumption of the potion is undesirable.
Movements after the liquid has been imbibed are inconsequential. Tossing or dropping the vial, spitting the cork, or anything.
Swallowing the vial whole, while possible (though painful), is not effective. The liquid needs to be absorbed by the body. Eating the vial like Ben did in Epic NPC man might lead to further injury. Both actions are forbidden.
Furthermore, waste is a concern. Getting 100% of the potion inside the body is very important and letting it drip down the corners of your mouth is a waste.
What's the optimal technique to drink potions fast?