There a item who have more chance ofmake the Red Rreign burn me for the achievement?

Any item with "Wreathed Red Rreign" in the name will get you or an inventory item (or maybe a pet? But I don't think I've seen that one yet this year) burned (but the inventory burning is broken right now, and doesn't actually burn an item). But no, there's no way to increase the chance that he'll burn you over your items or anything.