I think they must have meant the unvoiced "th" from "thing" not the voiced "th from "the". I've only ever heard "Anþony". I can't imagine any native English speaker saying "Anðony" unless they're drunk or otherwise slurring their speech significantly.
Out of curiosity, is it definitely the voiced TH from "the," rather than the unvoiced TH from "thing?"
In the US the spelling pronunciation of Anthony is by far the more common (at least in my experience), but I've never heard it with a voiced TH.