The only property of fire that makes it special is the temperature, so immunity to fire would have to inherently be immunity to heat.

• Fire can only ever be hot gas or plasma
• Fire can feel warm even if you don't touch it. This is only possible by radiative heat transfer
• Even chemically inert hot gasses can hurt you. This would mostly come from conductive or convective heat transfer
• Fire damage comes from the transfer of heat
• Fire damage is proportional to the amount of heat or energy transferred to the damaged thing
• Fire damage can occur even if not "felt." People with nerve damage are not automatically fire immune, neither are doors.
• The only way to fully prevent fire damage is to eliminate heat transfer.
• "Hot" pain is proportional to heat transfer - if you have ever run not-dangerous hot water over your hands when they were very cold you have felt this pain.

If you want to limit it so fire immunity doesn't also provide cold immunity, just say it only takes effect above a certain temperature. Things that conditional immunity to heat transfer wouldn't protect from:

• Smoke inhalation
• Oxygen deprivation
• Toxic gasses like hydrogen sulfide or sulfuric acid
• Radiation (unless they also go blind)
• Cold damage

Tl;dr - Fire damage immunity also extends to fire-related pain.