Nope, Mario has the widly popular "Super Mario Super Show", but that was a cartoon, not an Anime. It stretch first with the Super Mario Bros. 3 games before eventually going to Super Mario World and featuring Yoshi. But remember, it wasn't an Anime

The Zelda Series had a short-running cartoon series that was focused around the original Zelda/Zelda II games. It featured a horny Link always trying to score with Zelda, with the king laughing at Link's attempt to win his daughter over. And every once in awhile Ganon tries to take over Hyrule. Like Mario, it wasn't an Anime

Donkey Kong had that horrid 3d Animated "Donkey Kong Country", which throughlly disgraced the DKC franchise and DK himself with at least two songs every episode. Makes me vomit each time I think about it...thankfully not an Anime

Other than that, Kirby has "Hoshi no Kirby", better known as "Kirby: Right Back at Ya!", where it has all sorts of battles and such. Besides Pokemon, Kirby is the only major Nintendo franchise to have an Anime.

That's about it, so to recap...

Mario: Awesome Cartoon
Zelda: Awesome Cartoon featuring a horny Link
DK: HORRIBLE CARTOON
Kirby: Anime

Hope that helped