A spellcaster with 6 temporary hit points is hit for 3 damage. Do they have to make a concentration check?
According to 5e designer and official rules expert Jeremy Crawford, "When temporary hit points absorb damage for you, you're still taking damage, just not to your real hit points."
On 25 August 2014, Crawford was asked about a similar situation over twitter by @Mathew_Reuther:
@JeremyECrawford If my Temporary Hit Points are 10 and I take 30 damage from an attack, what is the DC of my Concentration Check?
@Mathew_Reuther The DC is 15. When temporary hit points absorb damage for you, you're still taking damage, just not to your real hit points.
For a Concentration check, "DC equals 10 or half the damage you take, whichever number is higher," so in the above twitter example, DC 15 is based on 30 damage. (PBR, p. 80; PHB, p. 203) While this situation is not exactly on point since the character took real damage on top of the temporary HP, Crawford's statement seems pretty clear that "you're still taking damage, just not to your real hit points."
However, if the actual damage was reduced altogether (not absorbed), then there would be no Concentration check. See, e.g., Deflect Missiles monk feature, Parry combat maneuver, and Heavy Armor Master feat. (PHB, pp. 74, 78 & 167)
Also, the abjurer's Arcane Ward feature has its own hit points1 and "[w]henever you take damage, the ward takes the damage instead." (PHB, p. 115) So, an attack that is wholly absorbed by the ward is not damage taken by the character and no Concentration check is needed. (Courtesy of @Dungarth)
- As opposed to giving a character temporary hit points like certain spells and magic items.
Yes, the concentration check is required even if you have temporary hit points.
A concentration check is required in any case the caster takes damage:
The following factors can break concentration:
Taking damage. Whenever you take damage while you are concentrating on a spell, you must make a Constitution saving throw to maintain your concentration.
There is no distinction about temporary HPs or normal HPs: the check is triggered by the damage.
This is confirmed by the Sage Advice Compendium:
If I have 10 temporary hit points and I take 30 damage from an attack while concentrating on a spell, what is the DC of the Constitution save to maintain my concentration?
The DC is 15 in that case. When temporary hit points absorb damage for you, you’re still taking damage, just not to your real hit points.
There is another entry in the SAC that indirectly confirms that even if you have temporary HPs you are still receiving damage (emphases mine):
How does Arcane Ward interact with temporary hit points and damage resistance that an abjurer might have? [...]
The ward takes damage first. Any leftover damage is taken by the wizard and goes through the following game elements in order: (1) any relevant damage immunity, (2) any relevant damage resistance, (3) any temporary hit points, and (4) real hit points.
Indeed, the wording above says that any leftover damage is taken by the wizard on the temporary HPs, if any, after having applied eventual immunities and/or resistances: the caster is hence receiving damage.
There is a answer provided by sadaqah based on rules intent. My answer is instead focused on the rules as written.
No, concentration checks trigger on taking damage, but temporary hit points are a buffer against damage.
Page 198 of the PHB says about temporary hit points:
Temporary hit points aren't actual hit points; they are a buffer against damage, a pool of hit points that protect you from injury.
Losing temporary hit points does not mean you take damage.