Except Wii Remotes weren't generally banned at Brawl tournaments, and every once in a while someone would actually use the Wiichuck set-up. Nintendo holds massive events with far more than 8 Wii Remotes and other wireless controllers operating in close proximity (like, say, their booths at E3). I seem to recall testing years back showing that hundreds of the things could be used in close proximity with no problems, basically suggesting that it's totally infeasible for Wii Remotes to ever interfere with each other since the number of Wiis you'd have to run in parallel to make that happen is completely impractical even for national events. It's true they can be interfered with especially if you have a ton of other wireless devices in play (we all remember the flubbed Skyward Sword demo), but I really doubt it will be a serious problem for tournaments.
A bigger issue is that controllers staying synced to consoles can result in interference in games of a different sort when the Home button is hit by somone not even involved in the match. The current tournament standard is a rule requiring battery removal when using such controllers, but that's hard to enforce and the scale of the issue will increase dramatically with everyone using such controllers. It's something we'll be able to deal with, but it will be a constant annoyance.
I have seen 36 wavebirds (yes, wavebirds) used in close proximity with no errors. All within the same cafeteria-size room.
There WILL be interference problems at some point, but I imagine the actual interference will be infrequent and small.
I won't get into the specifics of how wireless stuff works, but a simple "know this" list for TOs probably wouldn't be a bad idea.
Hell, I just realized we're going to have a lot of Wii U related problems in the tournament scene. I'll make a short list someone can steal right now.
1. Don't set two Wii Us right next to each other if at all possible. This increases the chances of interference.
2. Four people being synced to a Wii U can be a pain when singles starts. To fix this, use your Wii U gamepad to open up the menu and desync all available controllers. Restart the syncing process with only the controllers necessary; the other controllers shouldn't resync without manually pressing the lil red button.
3. The largest issue of interference will be due to several groups trying to sync simultaneously. To prevent potential conflicts, do things one at a time whenever possible. We may not have any trouble at all, but if this becomes a common issue it can be easily solved by simply calling out matches with a slight delay due to syncing. "Overswarm vs. Dazwa on TV 1, go ahead and sync. Ampharos vs. Mew2King on TV 2, wait until OS and Dazwa are done to sync" so on and so forth.
4. Battery life for controllers is going to be a large issue. Smashers being smashers will frequently forego charging their controllers prior to leaving for the tournament, often draining the batteries in a smashfest the day before. Have extra charge cables onhand and remind people to bring their own. Do not simply give out charge cables; take that person's license or have them pay for the cable in its entirety, and you give them back the money/license when they return the cable.
5. Wii U uses HDMI input only. All those old TVs don't work anymore. We will immediately have a shortage of TVs being brought to tournaments and many of those brought will be super small. Find venues that supply TVs for you to compensate for this early problem.
6. Wii U gamepads are expensive and will be needed to start the system. Have a charge cord of your own available as well as a Wii U gamepad of your own you can move aroudn to start systems that have been turned off. Expect raging to occur when people have their gamepad stolen or lost.
7. In reference to the above, LABEL YOUR WII U SYSTEM, GAMEPAD, AND CONTROLLER and encourage all others to do the same.
8. If you are having wireless issues, it could be any number of things. Try removing other wireless devices in the area to remove interference, including other Wii U consoles. This includes smart phones! "Please turn your smart phones off to reduce interference" could become a common reality.
9. Still having wireless issues? Try experimenting with some materials! Metal, for example, can block wireless communication. Walls can. Floors and ceilings. So if you're in a house, it might not be a bad idea to do some split level tournaments. Got a big venue with a bunch of tables and chairs? Take an extra table and put it up vertical between the systems. Depending on the material and the signal strength, it could help.
10. Wiimotes use the 2.4 GHz frequency. The Wii U tablet uses IEEE 802.11n at 5.2 GHz. I have no idea what the Wii U Pro controller uses. We will likely not have any trouble with the Wii U tablets as most things currently use the 2.4 GHz frequency. Wiimotes may have some trouble enmasse.
Probably lots more to use, I dunno.