Nintendo Germany has posted a video detailing how amiibo will be working in The Legend of Zelda: Twilight Princess HD. The Wolf Link and Midna figure will unlock the Cave of Shadows which is an arena-style dungeon pitting Wolf Link and Midna against an assemblage of monsters. When the trials are complete, the amiibo can be written to hold the amount of hearts a previous run had.
The Link and Toon Link amiibos restore the amount of arrows Link has in-game. While Zelda and Sheik figures restore hearts. Link, Toon Link, Zelda and Sheik can only be used once per day.
And for added challenge, the Gannondorf amiibo will make Link take double the damage for awhile.