No. Call of Duty: World at War runs entirely from disc. No installation required. Play Campaign, Multiplayer, Zombies on base Xbox 360 (4 GB internal / Arcade / Core) with zero extra storage.
Recommendation: Grab a 16 GB USB 3.0 stick ($5), configure it, install the game. Night-and-day difference in Zombies. If you play many 360 games, buy a used 250/320 GB HDD ($25–40).