RAW, no.
Or, answering the body question instead, the spell ignores these features.
Spells do (only) what they say. The spell does not force a saving throw. For the monster's feature to be a "Specific beats general" case, it should explicitly state something along the lines of:
If this creature is affected by the sleep spell, it gets a X saving throw. The consequences for failing are Y. The consequences for succeeding are Z.
Neither of them do, and, as far as I'm aware, no creature or class has a feature worded like that, i.e., no feature states that "they get a saving throw" in any way. They simply have advantage against saving throws that already exist, which sleep doesn't provide.
A similar case is
The mirror image spell has no effect on magic missile, which doesn't involve an attack.
So, we could answer with the same logic:
The bugbear chief's feature Heart of Hruggek has no effect on sleep, which doesn't involve a saving throw.
Additional conjecture for my answer is that you would have to adjudicate what type of saving throw? Is it Wis? Con? What happens if the creature succeeds the saving throw? Is it simply not affected? Does the HP from sleep's HP pool that would have been used for that monster get reduced anyway? While these answers might be easy to adjudicate, none of them is answered by neither the spell description or the feature. Spells are (supposed to be) clear (that's the whole point of "Spells do what they say"), so, if you have to adjudicate so many things, this is probably not how the effect should be resolved.
As for "Why does the Bugbear Chief have a feature like that then?", I've created this question. NautArch has already provided examples. One of them, the symbol spell, has a possible effect that states:
Sleep. Each target must make a Wisdom saving throw and falls unconscious for 10 minutes on a failed save. A creature awakens if it takes damage or if someone uses an action to shake or slap it awake.
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.
Best Answer
No
The Constitution saving throw to maintain concentration is caused by the damage - not the effect that caused the damage.
Similarly, your death saving throws wouldn’t have advantage if the damage that dropped you to 0hp came from one of these sources.