How do I set my Feedburner feed to use my favicon?

As @Rarst pointed out, there is no implementation of a favicon in the RSS spec (although there does seem to be one for Atom).

Feedburner (and other readers) rely on the de-facto standard that the favicon exists at the website root;

http://example.com/favicon.ico

Some clients may also try .png or .gif, but your safest bet is the standard .ico.