Many signs point to Feedburner closing soon as it's not being actively maintained anymore. Although it hasn't closed yet, users have complained about delays in email deliveries, and also loss of subscribers.
Therefore it might be wise to switch now.
Feedblitz has been around for a while, and they are a solid Feedburner alternative if you're looking for a service which sends out your posts automatically to your subscribers (based on your RSS feed), in a similar way as Feedburner does.
Feedblitz charges for the RSS- and Email parts:
See a full overview in Feedblitz' pricing table.
Switching from Feedburner to Feedblitz is made easy with a Feedburner migration guide.
Source: Feedburner-alternatives.com and Feedblitz.com
Feedpress is the to-go service if you are looking for a service to handle your RSS subscribers, esp. if you're a podcaster. Statistics are nice (see their features) and include Geolocation.
Feedpress also offers an RSS-to-Email solution, however you need to plug it into an external service.
Pricing: Feed analytics for 4 USD per month, podcasting solution for an additional 8 USD per month.
Source: Feedpress
SpecificFeeds is a free Feedburner alternative for both Email and RSS. It allows subscribers to filter the feed to only receive messages they want.
Source: Feedburner-Alterantives & SpecificFeeds
Feedio allows subscribers to follow not only certain blogs, but authors across many different blogs. They also position themselves as a possible replacement for Feedburner (however not tested yet).
If you don’t receive the email in the next 2 minutes please check your junk folder, and add [email protected] to your safe sender list.