What is the proper way to protect electrical wires in an iron pipe lamp?
Iron pipe is (aside from galvanizing and listing and some of the fittings) exactly the same as rigid conduit. Code does not require any protection for wires in conduit - indeed, code regards the conduit as protection for the wires.

All pipes should be de-burred on the inside at the cut and threaded ends (or cut and not threaded, depending how you are assembling) to prevent cutting the wire insulation. If you want to additionally use braided loom, that is fine, but not a code requirement.

If you wanted to be persnickety, you could use rigid (or IMC) conduit  and conduit fittings and then paint it black, rather than using black iron pipe.