Well now, the premise of the question no longer seems to be true after the latest errata, as you can't lose maximum hit points anymore through levelling. The following is my original answer, from when that wasn't true.
The rules don't make it completely clear
I suspect that there wasn't a lot of playtesting around the rules for creatures with 0 maximum hit points. As we'll see, there's some implied expectations that creatures only lose hit points due to damage, and it makes some rules come across awkwardly as one can also lose current hit points due to having one's maximum hit points decreased. Having one's maximum hit points go all the way down to 0 is something that's a consequence of other rules as you mention, but it's pretty unlikely to come up in practice.
But let's not let the rarity of this situation prevent us from exploring it. I'll give a bunch of quotes from the rules and tell you how I'm reading them. I'll mostly be quoting from the online Basic Rules, in the Combat chapter, in the Damage and Healing section.
What are Hit Points?
First, let's make sure we know what hit points actually are.
Hit points represent a combination of physical and mental durability, the will to live, and luck. Creatures with more hit points are more difficult to kill. Those with fewer hit points are more fragile.
A creature’s current hit points (usually just called hit points) can be any number from the creature’s hit point maximum down to 0. This number changes frequently as a creature takes damage or receives healing.
Whenever a creature takes damage, that damage is subtracted from its hit points. The loss of hit points has no effect on a creature’s capabilities until the creature drops to 0 hit points.
It's generally established, though I don't know of a rule that makes it perfectly explicit, that if your maximum hit points becomes lower than your current hit points, that your current points becomes your maximum hit points (See "What happens to current hp upon loss of max hp?".) Since due to the definition of "current hit points" it needs to be within the bounds of 0 to your maximum, it makes sense that it has to change, and it'd be real silly if it were to change to, say, 0, or to a random number less than the new maximum. Changing to the closest number that's still within the bounds allowed for current hit points makes the most sense, and nobody would really expect it to work differently.
Regardless, if a creature's hit point maximum is 0, then the creature's current hit points can be any number between 0 and 0, which I can't read as any other way than their current hit points become 0.
Does this change to 0 hp count as damage?
Your hit points just "becoming" a number is part of what's a bit odd about all this. Does this count as "damage", or is it just some other generic "change in hit point total"? As we'll see a little later, a lot of rules care about when damage is dealt, so this ends up being an important question.
Arguments in favor of it being damage
Arguments in favor of just being a change in current hit points
- Nothing in the rules calls out this reduction in current hit points due to a reduction in maximum hit points as "damage", and generally game terms are used in a somewhat consistent manner.
- The sentence "Whenever a creature takes damage, that damage is subtracted from its hit points." says that while damage subtracts hit points, it doesn't say that it's the only way to do so.
When describing how damage works,
Different attacks, damaging spells, and other harmful effects deal different types of damage.
None of the types listed seem to apply, as we have "rules damage" or "maximum hp reduced damage", and it's clearly a different sort of thing than the types of damage listed.
So which do I call it?
Well, feel free to read it however you find reasonable. In practice, we're going to get the most sensible answers if we call it damage, or at least if we decide to apply the same rules later on even though they say "damage" and it might not be in some technical sense. But if you're one who likes literal rules-as-written, if only as an exercise in how silly we can make these rules seem, there's certainly a good argument to be made that it's not damage and that damage rules shouldn't apply to this hit point change.
Do I die or just fall unconscious?
There's a rule that suggests one of those things sure ought to happen:
Dropping to 0 Hit Points
When you drop to 0 hit points, you either die outright or fall unconscious, as explained in the following sections.
This rule seems fairly clear. A creature at 0 hit points should either die outright, or just fall unconscious. So, which is it?
You don't die instantly
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.
And here's where you start to see why I made a big deal of whether of not this hit point change counts as "damage". This seems to only apply to damage that reduces you to 0 hit points. If your reading is that it's not damage, then this rule doesn't apply. But even if you say that the change in your hit points is damage, or that this rule should clearly apply anyway based on it being in the more general "Dropping to 0 Hit Points" section, I don't think there could be any "damage remaining". I can't see how this rule would cause you to die instantly.
You probably fall unconscious
If damage reduces you to 0 hit points and fails to kill you, you fall unconscious (see appendix A). This unconsciousness ends if you regain any hit points.
Now, this part really depends on whether or not you want to call the hit point change "damage", or apply the rule anyway (since it's in a more general "Dropping to 0 Hit Points" section) even if it isn't damage.
If you apply the rule, then you fall unconscious.
If you decide that it isn't damage and therefore don't apply the rule, then you need to treat that sentence starting the "Dropping to 0 Hit Points" section as just more an introduction to the section, and not really as a rule. Since clearly, even though that header text said that you would die instantly or fall unconscious, it appears that neither rule applied. There's also the sentence at the end of the Hit Points section "The loss of hit points has no effect on a creature’s capabilities until the creature drops to 0 hit points." You need to read that as saying that while a creature might experience some effect when dropping to 0 hit points, such an effect isn't required. If you're willing to take this approach to the text, which admittedly is a bit of a stretch, you'd be walking around completely fine even though you're at 0 hit points.
I imagine most playgroups would choose to apply the rule, and have the creature fall unconscious. But if your group has fun twisting rules in this way (and I've certainly had fun writing up this answer), I sure won't stop you from having an alert, awake, perfectly happy character who just so happens to be at zero hit points.
You make Death Saving Throws regardless
This part is actually pretty straightforward.
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.
Whether your group decided you fell unconscious or not, Death Saving Throws apply because you'd be starting your turn with 0 hit points and haven't become stable. And even if you roll a 20, you can't gain the 1 hit point that you normally would since your maximum is 0, though it does count as a death saving throw success.
If you do become stable, then your status doesn't change from there, since you can't regain the 1 hit point after 1d4 hours, either. So, you just remain unconscious (or remain conscious if your group goes that way), and you just stay that way since you have no way to get to 1 hit point unless some effect (probably caused by somebody else) manages to raise your hit point maximum back to a positive number.
Taking damage while at 0 maximum hit points is lethal
See this paragraph at the end of the "Death Saving Throws" section:
Damage at 0 Hit Points. If you take any damage while you have 0 hit points, you suffer a death saving throw failure. If the damage is from a critical hit, you suffer two failures instead. If the damage equals or exceeds your hit point maximum, you suffer instant death.
My reading of that last sentence is that if you take any positive amount of damage while your hit point maximum is 0, you will suffer instant death.