Re: nails in vintage tile hearth
Removing the nails will cause more damage than cutting and grinding them smooth. Sure, you're going to have nail spots that will rust over time, but if you don't want to damage the existing tile or replace it, then cutting the nails will be the better way to go.
Start by removing as much of the tack strip wood as possible. Use side cutters to nibble the majority of the wood tack strip away, then use the dremel with a cut-off wheel to snip the nails close to the floor, then install a sanding drum or grinding wheel to grind what's left of the nail smooth to the floor. I would recommend not trying to grind the first few nails flush until you've gotten the hang of using the dremel in this particular instance. As ed points out, they can get away from you pretty quick.
I suffer from CDO ... Its like OCD, but in alphabetical order, LIKE IT SHOULD BE!!!