Should I use plumbers tape on tub drain or just plumbers putty?
You don't need tape for the seal (those drain to shoe threads are not where the water seal takes place in those parts) but since you are using metal drain parts teflon tape or pipe dope does greatly increase the odds that you or the next person will be able to get them apart, rather than finding them irrevocably corroded together. I'd actually prefer pipe dope for the purpose.
As for putty, that apparently depends on the tub material. I was looking up instructions for my new-ish tub and was very surprised to find that silicone caulking was the recommended material for my tub - apparently plumber's putty does not get along with acrylic tubs? I'm going to guess that your all metal drain probably goes with a cast iron tub where plumbers putty is the right material, but I offer that observation because it surprised me, and made my tub of plumber's putty feel sad and unwanted, poor thing.