NO! But it can on the Game Boy systems but not on any ds systems.
The first generation of Pokemon games are incompatible with the Nintendo DS. If you are talking about Fire Red, which is the 3rd generation of Pokemon games and remakes of the original Pokemon Red, then yes. However that is no longer available for the DSi. Even the Gameboy Advance games are no longer compatible.
- 1st generation games: Pokemon Blue, Red, Yellow.
- 2nd generation games: Pokemon Silver, Gold and Crystal.
- 3rd generation games: Pokemon Ruby, Sapphire, Emerald, Fire Red, Leaf Green, Red Rescue Team and Blue Rescue Team.
- 4th generation games: Pokemon Pearl, Diamond, Platinum.
Yet to be released Heart Gold, Soul Silver are believed to fall under this category too.
That only covers the hand held Pokemon games. Sources Wikipedia