Burning Beetle is encountered as a red shadow in the second section of Caina within the Abyss of Time during The Answer.
Persona 4
Burning Beetles appear on the 6th and 7th floor of Yukiko's Castle in the Midnight Channel. Upon being defeated, they may drop Hard Horns. Selling 6 Hard Horns to Daidara will unlock the Shield Boots weapon for Chie Satonaka.
Despite not being set as weaknesses, Burning Beetles are twice as likely to be hit with Light/Dark skills and take double damage from Almighty attacks.
Golden
Burning Beetles appear on the fifth through seventh floors of Yukiko's Castle within the Midnight Channel. After being defeated, they may drop Iolites and Hard Horns. Selling 4 Hard Horns at Daidara Metalworks will unlock the Shield Boots weapon for Chie Satonaka.
Persona Q: Shadow of the Labyrinth
The Burning Beetle is a strong shadow that appears in Chapter 3 of You in Wonderland, and is the first of its kind of Shadows that rarely appear in the labyrinths. It also ambushes the party in P-Spots there. These Shadows have higher than usual HP, though not on par with F.O.Es, and are capable of attacking twice per turn.
As it is prone to doing massive damage to the party at this stage, proceed with caution. Equip sub-Personas with Tarunda to reduce the damage received from it, and pepper it with Ice attacks to kill it quickly. Like with many tough enemies, Poison can accelerate the pace of the fight greatly.