How to remove closet shelf clips and brackets without damaging the wall?
The wall is already damaged - nails do that to walls - so you're looking at fixing the wall no matter what.

With that in mind, you want a hammer, a small pry bar, a small quantity of joint compound or "wall repair spackle" (the latter is in the paint department), a sponge, a 1" putty knife, 1 pint or quart of primer and an equal quantity of interior paint matched to your existing paint as closely as possible, and a 1" trim paint brush.

Use the prybar and hammer to remove the clips and their nails.

Spread the joint compound smoothly over the damaged area with the putty knife, and allow to dry.

With the sponge DAMP NOT WET, smooth the compound even more.  Allow to dry again.

Apply primer with the brush to the dried joint compound.  Clean brush, allow it and the primer to dry.

Use same brush to paint the now dry white primer.