The ability holder cannot be frozen while the ability is active. If the ability is just gained or was ignored, the Pokémon will thaw their next turn. If it was frozen, it consumes the Aspear, Lum or Pumkin Berry it was holding before the ability activates. If Magma Armor is no longer the Pokémon's active ability, it does not thaw until it has switched back in.
Outside of Battle
If the ability holder is in the party, it reduces the number of cycles to hatch Eggs in the party by half. It shares this effect with Flame Body though, it does not stack with other Pokémon with this ability or Flame Body.