As of WordPress 3.7, there is an unregister_taxonomy_for_object_type
function available for just this sort of thing.
In your case:
// Remove tags support from posts
function myprefix_unregister_tags() {
unregister_taxonomy_for_object_type('post_tag', 'post');
}
add_action('init', 'myprefix_unregister_tags');
View the documentation for this function here.