Are Installed Xbox 360 Games Locked to One Console?
No. Xbox 360 game installations are tied to your Xbox Live profile (Gamertag), not the physical console.
How It Works
Disc Games: Install to HDD for faster loads/less noise. The disc must be in the tray to launch (license check). You can play on any Xbox 360 by signing in with your profile and inserting the disc.
Digital Games (Games on Demand / Arcade): Licensed to your Gamertag. Download on any Xbox 360 where you are signed in. Can also be played by other profiles on your Home Console (one console designated per account).
License Transfer: Move content licenses to a new console once every 4 months via Settings → Account → License Transfer.
Limitations
Only one console can be your "Home Xbox" for offline play by other profiles.
Simultaneous play on two consoles with the same digital license is blocked.
Disc installs do not make the game portable without the disc.
Your library follows your account, not the hardware.