The best replacement for feedburner is mailchimp.

The best alternatives to feedburner are rated as following:
★★★★★  mailchimp
★★★★  feedblitz
★★★  rss
★★  aweber
★★  google reader
★★  wordpress rss
★  jetpack
  subscribe2
  libsyn

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Google Trends

Scale is based on the average worldwide traffic of feedburner in all years. Click on graph for more.
feedburner
mailchimp
feedblitz