"Burning your feed" is just a cutesy way of saying "reroute your feed through feedburner"
I definitely think that using Feedburner (which is free) is a good thing for anyone to do, primarily because Feedburner allows you some basic tracking, so you can see how many people subscribe to your Feed and how many actually read it, which can be useful information.
Also, as Rebecca pointed out, you can use them to manage email subscribers as well, and it's completely automated - it picks up your feed and emails it to those on your email list (sometimes people prefer an email over an RSS feed reader)....this service is also free, HOWEVER, I do recommend that you give serious consideration to using Aweber instead IF you blog frequently.
The reason is that Feedburner is a daily service - if you publish a blog post, then at whatever time you've configured Feedburner to send to your email list, it goes out that day. And if you're blogging frequently - daily or several times a week - your emails will go out that frequently. For some people (me included), that's too much and I'll generally unsubscribe. Weekly is the most often I want to get email from anyone, no matter how valuable the topic, as my inbox is already more than I can manage.
Aweber on the other hand, allows you to specify how often you want the email to go out, so you can choose daily, weekly, monthy, or every XX number of days or blog posts. There is also much more customization you can do to the look of your email by choosing one of many templates.
The downside is that Aweber is NOT free - their cheapest plan is $20 monthly, but if your goal is to build a large mailing list, and use that list to generate revenue by promoting products, then it's worth considering.
SO burn your feed through Feedburner, but give some thought to whether to use Feedburner or Aweber to handle your email list.