Old horn is pretty brittle, and I would be afraid that I would crack it if leverage was applied by a nail puller. Better to just pull up on the tack if possible.
I think if I was going to try and remove and save the tacks, I would find a thin hardened steel wire...a strand from light aircraft cable maybe. Work it under the tack like dental floss, loop it around once and pull.....come to think of it, dental floss or fishing line might even work.
Disclaimer: YMMV...I have an old powderhorn to look at, but I have never tried what I just suggested on it.
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The man got the powr but the byrd got the wyng