This problem can be caused by requests for certain files that don't exist. For example, requests for files in wp-content/uploads/ where the file does not exist.
If this is the situation you're seeing, you can solve the problem by going to .htaccess and changing this line:
RewriteRule ^(wp-(content|admin|includes).*) $1 [L]
to:
RewriteRule ^(wp-(content|admin|includes).*) - [L]
The underlying issue is that the rule above triggers a rewrite to the exact same url with a slash in front and because there was a rewrite, the newly rewritten request goes back through the rules again and the same rule is triggered. By changing that line's "$1" to "-", no rewrite happens and so the rewriting process does not start over again with the same URL.
It's possible that there's a difference in how apache 2.2 and 2.4 handle this situation of only-difference-is-a-slash-in-front and that's why the default rules provided by WordPress aren't working perfectly.