The Legend of Zelda: Breath of the Wild will be receiving an amiibo range based on the game, these include Link as an archer, Link riding on a horse, and the guardian creature from the original trailer which will also be flexible which is awesome. Nintendo has made no mention of what these amiibo's will bring to the game, although I suspect arrow replenishment, and instant horse are pretty likely.
However Nintendo has detailed the games connection to the Wolf Link amiibo from The Legend of Zelda: Twilight Princess HD. Wolf Link can be summoned to help you in battle by attacking enemies on his own, he also will help you hunt which could be a major new mechanic, maybe eating is important. Wolf Link will have three hearts on a basic level, but I believe that he will have more based on your Twilight Princess data stored on the figure. Wolf Link will then disappear when his hearts run out. Here is the statement:
The game also includes compatibility with amiibo, which are sold separately. Nintendo announced a new series of amiibo figures specific to the game that includes Link with a bow (“Archer”), Link on horseback (“Rider”), and a “Guardian” that is the first amiibo with flexible parts. The existing Wolf Link amiibo works with the game as well: When players tap a Wolf Link amiibo to the Wii U GamePad controller, Wolf Link will attack enemies on his own and help players to hunt. He has three hearts, unless players carry over save data from The Legend of Zelda: Twilight Princess HD. Wolf Link disappears when his hearts run out.
You can also check out a trailer showing Wolf Link in action below:
You can also check out this other trailer that simply shows the amiibo's in all their glory: