In addition to
extension=fileinfo.so
I also had to add
mime_magic.magicfile=/usr/share/file/magic
to my php configuration. Then, I instantiate Finfo thus:
<?php
$mimeMagicFile = get_cfg_var('mime_magic.magicfile');
$finfo = new finfo(FILEINFO_MIME, $mimeMagicFile);
?>
(This is despite the rumours I hear about the default being exactly what I set it to anyway.)