Can I use Teflon tape to seal a flange?
In a word, no.  The plumber's putty is what forms the gasket between the drain insert and the bottom of the sink, and is likely what failed in the old one.  Telfon tape doesn't work very well as a seal unless it is forced into an extremely tight space (i.e. on threads) and is really only made of Teflon to reduce friction enough to allow it to stay in place long enough to get the load on it tight enough.