It's fair to say Nintendo's online efforts have been a bit behind-the-times. While Wii did support a number of online-connected games, Nintendo didn't offer much in terms of an integrated platform a la Xbox Live or PlayStation Network.
Wii U will be different, however. Already, a number of online-enabled games have been announced for the platform, including Call of Duty: Black Ops 2. The experience will be held together through the (still oddly mysterious) Nintendo Network. And like on PSN and Live, online games on Wii U will support voice chat... with a few caveats.
Whereas Xbox Live allows cross-game party chat, Nintendo Network will not. According to Kotaku, support for in-game voice chat will be enabled for "select games" only, meaning developers will have to enable the feature. (This is not unlike how it works on PS3.)
Oddly, although the Wii U GamePad has a microphone built into it, you will not be able to use it for voice chat during games. Instead, you'll have to pick up an officially licensed third-party headset from either Turtle Beach or Mad Catz (Nintendo does not appear to be working on an official first-party solution). These start at around $40. This means voice chat will be a pricier affair on Wii U; Xbox 360 includes a headset with the console, while PS3 is compatible with any Bluetooth headset.
You won't be able to use a Wii U headset with the Pro Controller either, as that peripheral does not include the necessary headphone jack to plug into. Instead, you'll need to plug the headset into the unused GamePad controller whilst playing with the Pro Controller. Cumbersome? Perhaps. But, dedicated Wii fans should be used to having all sorts of peripherals lying around when playing games.