How do I remove an IKEA shelf which is stuck on top of another?
Use the thinnest putty knife you can obtain to release the double stick tape. 

To get the remaining adhesive off, I would first use a dry rag and try to rub it off. Always rub in the same direction. If that doesn't do it, I would then use "Goof Off", alcohol or acetone to remove the adhesive left behind - first testing a hidden area with the product to make sure you won't effect the shelf laminate. You never know, most laminates are very durable, but...

Heating the adhesive can also work, but you need to have some care as the shelf laminate is also glued on to the wood product.

Hopefully, with care, you can get it all off just with the putty knife, but you may damage the laminate if you go at it too hard, so think of it as a 2 or 3 step process.  good luck!