What type of tape is appropriate for pipe insulation

One recommendation is to use rigid foam insulation attached with caulking, liquid nails, or silicone.

Cast iron. Which is conveniently already in the configuration you need at your local scrap yard: old radiators. That is, assuming this doesn't have to be potable.

Other than that I'd guess we're talking about some expensive exotic steel you probably can't wield very easily anyway, like lab equipment.

I'd like to think that regular old black pipe would stand up to this, especially if you could keep the fittings out of the flames and you mount it allowing for expansion.

However and whatever you build it out of, I'd test a mock-up before you start making endless loops only to find it pops your wields or the fittings. It's not the pipe walls, it's the joints. As long as they aren't subject to physical damage, your no where near hot enough to damage the metal.

Also... if you ever do get the pipes hot enough to melt with water inside them, something has blown up. What are we building here and where's the safety valve(s)? ;)