Check category before displaying featured image

has_category() will do the trick. You can pass it an ID or slug, or an array of IDs or slugs, and it will return true if the post has any of the given categories:

// ID
if ( has_category( 1 ) ) {
    the_post_thumbnail( 'full', array( 'class' => 'img-responsive' ) );
}

// Slug
if ( has_category( 'one' ) ) {
    the_post_thumbnail( 'full', array( 'class' => 'img-responsive' ) );
}

// IDs
if ( has_category( [1, 2, 3] ) ) {
    the_post_thumbnail( 'full', array( 'class' => 'img-responsive' ) );
}

// Slugs
if ( has_category( ['one', 'two', 'three'] ) ) {
    the_post_thumbnail( 'full', array( 'class' => 'img-responsive' ) );
}

And you can check if the post is not in certain categories by adding ! to the condition, to indicate ‘is not’:

if ( ! has_category( [1, 2, 3] ) ) {
    the_post_thumbnail( 'full', array( 'class' => 'img-responsive' ) );
}