The service will remain - but
with no API, no one will receive your blog posts through your feedburner RSS.
So, if a blog has 1000 RSS subscribers via feedburner on 19th October, they will have none on 20th October.
Equally, any services that people use to share your work with their followers via your feedburner RSS will cease to work.
The carrier here, is that it's possible
Google will offer some kind of redirect - however, as Neville says in his post, there has been no word from Google regarding this.