Moved my WordPress installation to a new server. I kept all paths etc intact, copied my VirtualHost and permissions were good, so I was at first surprised when my images didn't show and redirects didn't work. The cause was quickly found, however; it seems my old install had different default settings hidden somewhere in the Apache configuration.
I guess there may also be different causes, but in my case I fixed it by adding this to my Wordpress VirtualHost:
Allow from all
/var/www/wordpress is obviously the DocumentRoot of my WordPress install.
Lo and behold, my WordPress site wasn't broken anymore.