According to Bulbapedia, there are four glitches in both Pokemon Fire Red and Pokemon Leaf Green. These inlcude the Infinite Nugget, the Poke Flute, the Roaming Roar and the Low IV. These glitches differ from Generation I's Red, Green and Blue.
The first glitch is the Infinite Nugget glitch. This is technically not a glitch, but it is usually considered one. To receive infinite nuggets, the player needs to lose and replay the Rocket Grunt leader at the end of Nugget Bridge. Every time the player loses, the Rocket Grunt gives the player a nugget. The Grunt can play played infinite times, meaning the player can receive infinite nuggets.
The second glitch is the Poke Flute glitch. This glitch allows the player to mute the Poke Flute music when played. To perform this, simply activate and deactivate the help menu while the Poke Flute is playing.
The third glitch is the Roaming Roar glitch. This is a glitch which makes any roaming Pokemon disappear from the entire game if the Pokemon uses Roar during battle.
The final glitch in Generation III is the low IV glitch. This glitch is the result of missing code. It gives roaming Pokemon low IV levels which, while not affecting HP, will lower all other stats such as attack and defense. This is unfortunate for IV trainers who wish to maximize a Pokemon's abilities. Glitches found in Generation I, such as the Missing No. glitch, have been removed from the remakes.