A PC is adjacent to a target which is a Magma Mephit. The mephit does damage to adjacent creatures upon death:
Death Burst: When the mephit dies, it explodes in a burst of lava. Each creature within 5 ft. of it must make a DC 11 Dexterity saving throw, taking 7 (2d6) fire damage on a failed save, or half as much damage on a successful one.
The PC has just cast the Thunderwave spell, targeting the mephit in the spell's area. This spell states:
Each creature in a 15-foot cube originating from you must make a Constitution saving throw. On a failed save, a creature takes 2d8 thunder damage and is pushed 10 feet away from you. ... In addition, unsecured objects that are completely within the area of effect are automatically pushed 10 feet away from you by the spell’s effect
The mephit failed the saving throw and took enough damage to kill it. The spell's description doesn't specify whether this would occur before it moved, or not.
The question Does Thunderwave's push effect knock away incoming spells/projectiles? is somewhat related, but the answer relies mostly in the instantaneous nature of the spells and the lack of the word "object" in the spells.
I think it's a reasonable DM judgment that the exploding bits of lava count as "unsecured objects" and that no damage would occur, but there are other impacts: another mephit is 10 feet behind it and if the explosion occurred later, that mephit would take damage from the explosion. Given that the mephits are concentrating on the Heat Metal spell (on the PC's weapon and armor), any damage requires a constitution saving throw which significantly impacts the PC's fighting ability.
Does its Death Burst do damage to the attacker, or was it pushed more than 5 feet away before its explosive demise, and may damage someone close to its new location?