Death, if it is "just a monster"
As the "Monsters and Death" section of the PHB explains:
Most DMs have a monster die the instant it drops to 0 hit points, rather than having it fall unconscious and make death saving throws.
Mighty villains and special nonplayer characters are common exceptions; the DM might have them fall unconscious and follow the same rules as player characters.
If the creature is narratively important, it will "follow the same rules as player characters", which are more complicated.
Otherwise, it can continue to act at 0hp
Player character rules are contained in the larger Dropping to 0 Hit Points unit. This begins:
When you drop to 0 hit points, you either die outright or fall unconscious, as explained in the following sections.
As the introduction to this unit, this is a general statement that these are the typical consequences of falling to 0hp - either death or unconsciousness. However, nothing here says that if you can't be affected by one, you must immediately have the other. Rather, this introduction is saying to apply the "following sections" to determine what happens. The "following sections" are (1) Instant Death, (2) Falling Unconscious, and (3) Death Saving Throws. They are meant to be applied in order.
Unless the blow that took you to 0hp did exactly enough damage to place you at 0, you have damage 'left over'. If this left over damage is equal to or greater than your maximum hp, you die immediately. If this does not apply, proceed to section (2).
If you did not immediately die from the blow that brought you to 0, you are now unconscious. Proceed to section (3). If you are Immune to the Unconscious Condition, you would not be unconscious - but you would proceed to (3) all the same.
This is the longest section, and explains how to make death saving throws.
Whenever you start your turn with 0 hit points, you must make a special saving throw, called a death saving throw, to determine whether you creep closer to death or hang onto life.
Notice that unconsciousness is not a precondition for section (3). You make death saving throws whenever you start your turn at 0 hp (and later we learn you also have a failed save applied whenever you take damage while you are at 0hp) - regardless of whether or not you are unconscious.
Thus, for a monster like an elemental with immunity to unconsciousness, and for which the DM wished to use the PC rules:
(1) The DM would first check to see whether the elemental was killed from massive damage. If not, section (2) would be ignored due to the monster's immunity1. Finally in (3) we would have a monster conscious and capable of action at 0hp. The monster would need to make a death save at the start of every turn, and would take a failed save every time it subsequently took damage, but otherwise it could keep acting at zero hp until it either died from failed saves, or was healed or spontaneously recovered. It would certainly not be incapacitated, as nowhere in the rules for operating at 0hp does it say that this causes incapacitation.
1And see also: Is an Immune creature considered to have the condition without suffering its effects?