Filtering out specific post formats from certain pages

Question

Since you’re going to be modifying either the Loop Query or the Loop output, I would recommend modifying the appropriate template files directly.

To filter out Posts with the “quote” Post Format, simply wrap your Loop output (inside of the if-have-posts-while-have-posts-the-post) with:

if ( ! has_post_format( 'quote' ) ) {
    // Loop output goes here
}

Then, in your sidebar, you would create a custom Loop, e.g.:

$linkpostargs = array(
    'tax_query' => array(
        'taxonomy' => 'post_format',
        'field'    => 'slug',
        'terms'    => array( 'post-format-link' ),
        'operator' => 'IN'
    )
);
$link_posts = get_posts( $linkpostargs );

foreach ( $link_posts as $link_post ) {
    // Custom Loop output goes here
}

(H/T Michael Fields for the query args)

I honestly don’t know how you’d replicate this functionality using a Plugin alone.