Yes Google did officially shut down access to Feedburner's APIs back in May 2011.
It does still work yes, you can still use it to burn your feed with them.
Having a feed or RSS sitemap is just one part of on-page SEO just like having an XML and/or HTML sitemap is.
Your feed is used for off-page SEO purposes too such as for providing extra reading options, RSS feed readers, aggregators, RSS email subscriptions etc.
FeedBurner just provides these features and others like email subscribing, podcasting, feedflare, buzzboost, pingshot, feedcount chicklet etc
But it just means that if you use/used Feedburner as a service that enables readers of your blog to receive your content via RSS or email every time you update the blog that will cease working.
But there are a few other FeedBurner alternatives too like like FeedBlitz, Feedcat, IFTTT, Aweber or Mailchimp.
A lot of people moved to FeedBlitz after the shutdown.
