The rules for Feedburner should come before the WordPress rules. And activate the rewrite engine just once. That’s enough. 🙂
<IfModule mod_rewrite.c>
RewriteEngine On
RewriteBase /
RewriteCond %{HTTP_USER_AGENT} !FeedBurner [NC]
RewriteCond %{HTTP_USER_AGENT} !FeedValidator [NC]
RewriteRule ^feed/?([_0-9a-z-]+)?/?$ http://feeds.feedburner.com/ZoomingJapan [R=302,NC,L]
RewriteRule ^index\.php$ - [L]
RewriteCond %{REQUEST_FILENAME} !-f
RewriteCond %{REQUEST_FILENAME} !-d
RewriteRule . /index.php [L]
</IfModule>