Echo the custom field value from the latest post from the ‘video’ post format?

While this may not answer your question, but it will be helpful to solve it. It seems like the statement

 $video = get_post_meta($postid, 'dt_video', true); 

is assigning blank string value to $video variable and thus iframe code becomes

<iframe width="180" height="101" src="https://wordpress.stackexchange.com/questions/139255/?rel=0" frameborder="0" allowfullscreen></iframe>

which is the cause of your site being displayed in the iframe.

And for your question:

I want to show the latest video posted which is decided by the latest post with the video url custom field filled in

I would suggest you to paste your code for selecting posts from db throug query, so I can have insight.

Edit:

change your query args parameter to exclude blank values in dt_video in post meta field.

$args = array( 
    'numberposts' => '1', 
    'tax_query' => array(
        array(
            'taxonomy' => 'post_format',
            'field' => 'slug',
            'terms' => 'post-format-video'
        )
    ),
    'meta_query' => array(
        array(
            'key' => 'dt_video',
            'value' => '',
            'compare' => '!='
        )
    )
);

check WordPress documentation for more information :http://codex.wordpress.org/Class_Reference/WP_Query#Custom_Field_Parameters