Yes. An active Xbox Live Gold subscription (now Xbox Game Pass Core) is required to play Halo 3 online multiplayer on Xbox 360.
The single-player campaign, local split-screen, and System Link (LAN) multiplayer work without a subscription. However, all matchmaking, ranked/social playlists, co-op campaign over Live, and Forge/Custom Games browser require Gold.
Note: On Xbox One/Series X|S via backward compatibility, the same requirement applies—Xbox Game Pass Core or Ultimate is needed for online multiplayer.