The Fire Curse is a name for the trend that TwitchPlaysPokemon never has a Fire-type in the Hall of Fame.
The Fire Curse was first noted when Abby was released, then again with the release of Flareon. At that time, it was believed that the curse simply was that Fire-type starters must be released. After the events of Bloody Sunday, the Curse shifted to say all Fire-types must be released.
In Crystal, the Mob captured a Ponyta that survived until the end of the game. However, this Ponyta never joined the party, instead spending the whole time in the PC. It was then believed that either Fire-types must join the party to be released, or that the Curse simply prevents them from joining the Hall of Fame.
It was considered the reason that Zexy got released in Emerald.
Multiple Fire-types were captured in FireRed, but none of them made it into the Hall of Fame, establishing the Curse as meaning Fire-types cannot enter the Hall of Fame. Flamesplash is exempt due to not having Fire as a type, just as moves and ability.
The Curse struck again in Platinum when Fire-type starter Chimchar was released less than one day into the run. However, the Curse was believed to have finally ended in Platinum, when Solareon the Flareon entered the Hall of Fame as the party's highest-leveled Pokemon.
The curse continues to some extent to this day, with Sarfa in Omega Ruby and Growleerzard in Anniversary Red both being released. FFF, the Charizard fire starter of Moemon, survived though and reached the Hall of Fame, while R-Ceus, the Torchic starter of Nina Q, also survived but did not enter the Hall of Fame. BEST of the Vietnamese Crystal run also survived the curse, due to the game being played purely in Democracy mode.