What your basically talking about is creating a web proxy for the feed, and then serving the proxied feed to your users. By default this method will not update the feedburner subscriber counts as they will only see 1 unique visitor requesting the feed, however there are methods to inform google that you are requesting the feed on behalf of a certain number of subscribers.
In the case of stand-alone feed readers, that user has an application
running on their computer which fetches the feed repeatedly throughout
the day. We look at characteristics of those requests, and
differentiate between repeated requests from the same person (as
indicated by regular polling intervals, consistent IP addresses, and
common user agents) and different requests (where one or more of the
previous data points vary). In the case of web-based feed readers (My
Yahoo, Google Reader, Bloglines, Pageflakes, etc.), those services
retrieve the feed repeatedly throughout the day, but do so on behalf
of multiple people. Almost all of these services report to us how many
of their users are subscribed to the feed. At the end of the day, we
tally up how many stand-alone feed readers are subscribed, and add
them to the web-based users. The end result is the total subscriber
number we report. (I’m leaving a few details out; see below for a more
complete answer.)
from here