No. The Wii functions fully offline: plays discs, saves to internal memory/SD, uses Wii Menu, Photo Channel, Mii Channel, and disc-based games without any connection.
Note: Official Nintendo Wi-Fi Connection (WFC) for online multiplayer/shop **shut down May 20, 2014**. The below describes historical functionality or homebrew-restored services.
Bottom Line: Internet not required for core experience. Today, it's primarily for homebrew/online revival. If you don't homebrew, you only need it for the initial 4.3 update (if not already on it) or Netflix (if app still works on your region/model).