Retrieve value of a category’s custom field

http://www.advancedcustomfields.com/resources/get-values-from-a-taxonomy-term/ prety much covers what you’re after I believe.

So in your example

while ( $projects->have_posts() ) {
    $projects->the_post();

    $category_classes = array();
    $categories = get_the_terms( get_the_ID(), 'project_category' );
    if ( $categories ) {
        foreach ( $categories as $category ) {
            $categories_included[] = $category->term_id;
            $color = get_field('color', 'project_category_'.$category->term_id);
        }
    }

You can also pass a term to get_field and the_field, so you could also use the following depending on your version of ACF as I believe much older versions don’t support passing terms

$color = get_field('color', $category);