What kind of adhesive would I need to install ceramic tile on to the inside backing of a bookcase?

 

How big is the tile? Assuming the backer is substantial enough and will not flex, which is often not the case. If it flexes at all, the tile will pop off.

 

I would use silicone or construction adhesive like PL200/400 or Liquid Nails. This assumes you will not be installing grout.

 

In a cabinet I'd use construction adhesive. Then if you want to "grout" I would not use ordinary grout but instead would use colored silicone caulk. If it's an intricate grouting job a rag soaked in rubbing alcohol (I like to use 70% or stronger) will allow you to work the silicone like grout without is smearing and sticking badly.