Adding an array from a query string to a WP meta_query

This $_GET['variable'] is an array, as per your URL query string ?variable[]=value1&variable[]=value2, $_GET['variable'][0] and $_GET['variable'][1] should return those 2 key values

Edit – after the discussion – making it dynamic

$meta_query = array();
if ( ! empty( $_GET["variable"] ) ) {
    if ( is_array( $_GET["variable"] ) ) {
        $meta_query['relation'] = 'OR';
        foreach ( $_GET["variable"] as $value ) {
            $meta_query[] = array(
                'key' => 'my_post_field',
                'value' => sanitize_text_field( (string) $value ),
                'compare' => '='
            );
        }
    } else {
        $meta_query = array(
            'key' => 'my_post_field',
            'value' => sanitize_text_field( (string) $_GET["variable"] ),
            'compare' => '='
        );
    }
}
$filter = array(
    'post_type' => 'my_custom_post_type',
    'meta_query' => array(
        'relation' => 'OR',
        $meta_query
    )
);
$posts = new WP_Query( $filter );

I have changed the operator to = because I was testing it and it worked on my local installation.

meta_compare (string) – Operator to test the ‘meta_value’. Possible
values are ‘=’, ‘!=’, ‘>’, ‘>=’, ‘<‘, ‘<=’, ‘LIKE’, ‘NOT LIKE’, ‘IN’,
‘NOT IN’, ‘BETWEEN’, ‘NOT BETWEEN’, ‘NOT EXISTS’, ‘REGEXP’, ‘NOT
REGEXP’ or ‘RLIKE’. Default value is ‘=’.
— codex