Does Mind Sliver’s saving throw penalty apply to Concentration saves from its own damage

No saving throw is triggered

I found two relevant Q&A twitter posts from Jeremy Crawford (the Lead Designer and Managing Editor for DnD 5e) that shed some light on this. However, Jeremy's tweets are no longer considered official, but they can provide some guidance into his thought process at the time he wrote the tweet.

Question:

if a raging barbarian makes no attack, but takes one damage that is reduced to zero, does rage drop?

Answer:

Taking 0 damage is the same as taking no damage.

If you took no damage, you didn't take any damage.

Taking 0 damage does not count as taking damage, therefore it shouldn't trigger any effect conditioned upon taking damage.

I also found a specific example of a different effect that triggers on damage (in this case, drow poison from the DMG):

Question:

If the Battlemaster maneuver parry, reduce the weapon Drow damage to zero, the poison damage still work ?

Answer:

Drow poison in the DMG is delivered by piercing/slashing damage (0 dmg = 0 poison).

To me, this lays out a clear line of logic. Dealing damage applies Drow poison. If you deal 0 damage, it does not apply the poison according to Jeremy Crawford. Therefore, dealing 0 damage does not count as dealing damage for effects that trigger on dealing/taking damage. This would logically include concentration checks.

Officially, the Sage Advice Compendium document states that Supernatural Defense does help with maintaining concentration

Q. Does a Monster Slayer ranger’s Supernatural Defense feature apply if a creature damages the ranger, thus causing the ranger to make a Constitution saving throw to maintain concentration on a spell?

A. Yes.

I am somewhat saddened by this response; it is quite unlike the SAC doc to simply state an answer without any sort of an explanation as to why that is the answer. So while this is the official ruling and is likely also how the feature was intended to work, I do not find the evidence they provided (none) particularly convincing. Below is my original answer to this question, from the evidence I could find:

Personally, I believe that Supernatural Defense does not help with maintaining concentration

From the question "When must the wizard choose to overchannel?", we know that if something says "When you cast a Wizard spell... that deals damage" it does not mean you wait to see if the spell deals damage; you decide to use Overchannel immediately when casting the spell.

Thus we must know whether a spell deals damage before its effects take place (otherwise how would we know what spells are eligible for Overchannel?); the only way to know this is the spell's description. Therefore, a spell that doesn't deal damage normally but happens to move a creature onto a damaging area does not suddenly count as "a spell that deals damage". Whether such an area existed is outside of the spell's control.
What is forcing you to take damage is not the spell, but the area of effect.

The Sorcerer's Careful Spell Metamagic uses similar wording, stating:

When you cast a spell that forces other creatures to make a saving throw, you can protect some of those creatures from the spell’s full force...

Similarly then, we must know whether a spell forces a saving throw before its effects take place (otherwise how would we know what spells are eligible for Careful Spell); the only way to know this is the spell's description. Therefore, a spell that doesn't cause a saving throw normally but happens to damage a creature that is concentrating on a spell does not suddenly count as "a spell that forces a saving throw". Whether the target was concentrating is outside of the spell's control.
What is forcing you to make the saving throw is not the spell, but the damage.

This also prevents the unusual scenario of a Sorcerer using Careful Spell on a spell like firebolt. If the target were concentrating on a spell, and firebolt counted as forcing them to make a saving throw, then the Sorcerer could use Careful Spell but it would have no effect whatsoever. Though this isn't necessarily disallowed, it is exceptionally odd and points towards saving throws to maintain concentration not counting as having been forced by the spell.

I don't see any greater link between a spell happening to move a creature onto a damaging area and a spell happening to damage a creature that is concentrating on a spell (neither results are under the spell's control as they rely on outside forces). So if a spell resulting in a saving throw to maintain concentration counted as "a spell that forces a saving throw" then a spell which moves a creature onto a damaging area would count as "a spell that deals damage", which is almost certainly incorrect.

I believe that a spell only forces a creature to make a saving throw if the saving throw is part of the spell's description, not if it just happens to result in a saving throw being made.

All of this argument extends to the Hunter feature:
A creature moving you onto a damaging area of effect would not count as them forcing you to take damage and a creature damaging you would not count as them forcing you to make a saving throw to maintain concentration. It was not in their control whether the damaging area exists or whether you are concentrating on a spell.