Update: Bethesda noted that if you do decide to install Doom 3 BFG, the previous games will still be available from the Xbox 360 dashboard, just not from the in-game menu. Here is the official response:
"If you do install DOOM 3 BFG Edition to your Hard Drive, you will still be able to access both titles from the Xbox LIVE Dashboard under 'My Games'. If you do not already own XBLA versions of Doom 1 and/or Doom 2, you will need to have the BFG game disc in the 360 to launch these titles from 'My Games.'"
Original Story: In a bizarre technical hitch, installing Doom 3 BFG onto the Xbox 360's hard drive makes the original Doom and its immediate sequel unplayable.
Twitter user Wario64 captured the error message for all to see.
Bethesda's global community lead Matt Grandstaff confirmed to NowGamer, "If you choose to install the game to the Xbox 360 hard drive, you will not be able to launch Doom or Doom 2."
Furthermore, he noted, "As there is no performance advantage to installing the game to your hard drive, we recommend not installing the game to avoid this issue."
Generally installing a game helps speed up load times, so it's very unusual that doing so has no benefit here. But hey, at least this means you can bypass the arduous installation process without feeling like you're missing out on anything.