How to attach metal knife rack to tile without ruining tile?
If the tiles have a non-porous glaze, an adhesive over a larger contact area should work fine.
Adhesive Joint Strength and Surface Area
The strength of the adhesive joint is proportional to the area in contact.  This is how drywallers hang 100lb sheets of drywall with adhesive and almost no screws: Because of the large area over which the adhesive spreads the holding force to the studs or joists.  The larger the sheet of drywall, the larger the adhesive surface area, the higher the holding strength.
Adapter Plate
Because the rack's mounts have small surface area, you would build an adapter plate with adequate adhesive contact surface area to support the weight.  You mount the rack to the plate and adhere the plate to the tile.
Suppose the plate had an area 4&quot;x4&quot; in contact with tile.  That bond will be 20x the strength of the 0.9&quot;x0.9&quot; area of your existing mounts.  Increasing the contact area is how you get strength from your adhesive joint.
Adhesive Type
Once you've increased the contact surface area, you are no longer requiring anything extraordinary from your adhesive.  I would guess that something like silicon tiling caulk that is designed for clean removal from tile would work best.  You don't want something with too much holding force per unit area because it will take too much force to pop the plate off when you're done.
Plate Material
Ideally the plate material is something slightly flexible so that you can peel rather than pop the adhesive joint when you remove it.
If the plate is rigid then you'll be applying the entire failure force to the tile when you remove it.  This could crack or lift the tile.
On the other hand, a flexible plate allows you to peel the plate which means overcoming only a small area of holding strength at a time.  This is the exact same principle as peeling scotch tape.
Sacrificial Test
To be safe you'll want to test a representative mockup of an adapter plate and tile to test the following:

the plate/tile bond is adequate to support the weight
the adhesive can be removed without marking or marring the tile or grout
you can pop the plate from the tile without breaking or lifting the tile