When moving your blog, making sure everyone finds you is super important. Especially if you have an existing following.
Even if you are keeping your URL you still need to make sure that your new site is feeding your old feedburner set-up.
Here's how you do that.
Step 1: Login to Feeburner. (http://feedburner.google.com)
Step 2: Click on the blog/site that you'd like to edit.
Step 3: At the very top you'll see a link called "edit feed details..." click that bad boy.
Step 4: Delete your old original feed with your new one. If you have a WordPress blog the default is http://yourURL.com/feed
Step 5: Click "save feed details"
That's it! It should take you 5 minutes or less.
Note: It might take up to 24 hours for your RSS feed to start showing up again. And it will dump all of your current RSS feed into people's RSS reader of choice. (Defined by your WordPress settings)