How do I install a vanity that's higher than the mirror?
Is the mirror one piece or is it made of tiles?
Is it glued in place or supported by clips, brackets, or a frame?

If it is mounted (clips, brackets, or frame) carefully unmount it, have it professionally cut, and then remount it.  I strongly advise hiring a professional window or glass man to do this. 

If it is glued but is made of tiles:


Locate a supply of identical tiles.  
Have enough of the new tiles cut down and be sure to include some spares.
Carefully remove the tiles that will be partially blocked by the new cabinet.  
Carefully glue the cut-down tiles in place.



If it is glued and is in one piece:


Do not try to cut it yourself!
If you find a glass man that claims he can cut it, be sure he is bonded and contracts to fully replace the mirror if he damages it.
Otherwise, leave the mirror uncut and...
Use sheetrock, plywood, a laminate, etc. to shim the cabinet out to where it is at least 1/16th away from touching the mirror.
Carefully caulk, frame, and/or trim for the best aesthetics.  Be sure that there is a water-tight seal to stop moisture on the mirror from dripping behind the cabinet.