The rules on dropping to 0 hit points (PHB p. 197) almost entirely depend on taking damage. The section begins as follows:
When you drop to 0 hit points, you either die outright or fall unconscious, as explained in the following sections.
However, we then have the two following sections which make reference to dropping to 0 hit points:
Instant Death
Massive damage can kill you instantly. When damage reduces you to 0 hit points and there is damage remaining, you die if the remaining damage equals or exceeds your hit point maximum.
Falling Unconscious
If damage reduces you to 0 hit points and fails to kill you, you fall unconscious (see the appendix). This unconsciousness ends if you regain any hit points.
In fact, the only reference which doesn't include the condition of damage pertains to monsters:
Monsters and Death
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.
The only rule pertaining to player characters which doesn't involve damage is what happens after one is at 0 hit points:
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.
All emphasis mine. All of this raises the following question: in the absence of assuming intent (however clear it might seem), are there any rules for determining what happens to a player character who drops to 0 hit points as the result of an event which does not involve damage, such as the expiration of the aid spell?
Best Answer
The creature either dies, or falls unconscious
The rules say (PHB p. 197):
Dying outright
When does the creature die outright? In two cases: either due to massive damage, or because the creature is a monster and the DM does not want to make death saving throws for it as it is too much of a hassle that does not contribute to gameplay.
Falling unconscious
In all other cases, the creature does not die outright. It logically follows that the creature then falls unconscious. There are only these two options, to die outright or fall unconscious.
The rules confirm this happens in case of damage reducing you to 0 hit points explicitly, and generally:
This only leaves the case if you are reduced to 0 hit points from sources other than damage (like the dropping of of Aid). Do you fall unconscious or die?
This case is not covered explictly and will require the DM to make a judgment call. The treatment for damage sources suggests that unconsciousness is intended as the default effect of being at 0 hit points, as it indicates there needs to be something special about the damage to kill you. It would stand to reason that the same logic, of non damage sources reducing your hit points likewise require something special to kill you, but it is not airtight.
Once unconscious, the creature has to start making death saving throws on subsequent turns, and then will either stabilize, on a lucky 20 awake with 1 hit point, or die.
Special cases
There are a few special effects that lead to other outcomes when you fall to 0 hit points, for example, disintegrate:
All these cases are applications of the specific beats general rule, where a more specific rule overrides the general rule of either dying of falling unconscious.