I have tested several games. Keep in mind I'm currently testing them, this will be updated as I test every games.
Here are the current list of working/not working online games in Wii, without using custom servers :
NOTE : Disabled online games have the error code : 20110
Still Working :
Call of Duty : World At War (And possibly other CoD games as well)
Castlevania Judgement
Dragon Ball Z Budokai Tenkaichi 3
Final Fantasy Fables Chocobo's Dungeon (CAPTURED)
Goldeneye 007 (CAPTURED)
Gunblade NY and LA Machineguns : Rage of the Machines (Leaderboards only game)
Naruto Clash of Ninja Revolution 3
Pro Evolution Soccer 2013
Rayman Raving Rabbids : TV Party
Rayman Raving Rabbids 2 (Leaderboards only game)
Raving Rabbids : Travel in Time
Tatsunoko VS. Capcom - Ultimate All Stars
Tom Clancy's Ghost Recon (Leaderboards only game)
Yu-Gi-Oh 5D's Duel Transer
Not Working :
Animal Crossing : City Folk
Conduit 2
Dr. Mario Online RX
Mario Kart Wii (Captured)
Samurai Warriors 3
Sin and Punishment 2 : Star Successor
Super Smash Bros Brawl
Tetris Party Deluxe
The Last Story
Virtua Tennis 4
Wii Chess
Wii Music
NOTE : Needless to say, first party Wii Games' online features are disabled.
Unsure :
Big Brain Academy : Wii Degree
WiiWare Games
You can still save the working games, but keep in mind it might be shut down soon.
As said in Chocobo's Dungeon official Nintendo page, the WFC has been disabled since the 20th May, but I managed to connect it to WFC and quickly captured it with barronwaffles, lol.
Also, my speculation is Ubisoft, Konami, Bandai Namco, Activision, EA, and Capcom are using their own servers.