The Feed Manager is trying to make a web request from server side code to get the feed but it is failing. In this case the remote server is the web server but it needs to retrieve the feed over the internet even though it is the same machine.

"System.Net.WebException: Unable to connect to the remote server"

This can happen in Medium Trust hosting, some medium trust environments allow server side web requests and some don't. I know that when testing medium trust on my local machine it works when medium trust is configured as follows:

<trust level="Medium" originUrl=".*" />

the bold part is what is needed to allow server side web requests to work in medium trust. However in a hosting environment only your host could change that if they are enforcing medium trust so you will have to get help from them to resolve it.

If you look under Administration > System Information and it says "Partial Trust" that means you are hosted in medium trust.

ok, then either the feed url is bad or it is not accessible to the web server for some reason. could be a number of possible causes like  firewall restrictions or the dns is resolving to the wrong ip address or other network problems.