Mounting a hair dryer holder to a tile wall
In order to mount in on a tiled wall you need to drill a hole in the tile without cracking the tile. 

I use a diamond encrusted hole saw and water like this one. 



Go slow and you will have clean holes you can then insert  anchors into. 

This type of anchor should be sufficient since you have tile on the wall providing rigidity.

( Sized appropriately for the hole size you create. )

If you are lucky, or plan it, you may have a stud where you are mounting the holder. If so then you do not need the anchors, just screw into the stud.