Inserting HTML tag with ACF into shortcode

You have to use do_shortcode() to execute shortcodes in your string

Full code should be like that

<?php
function my_shortcode( $atts ) {
    $atts = shortcode_atts( array(
        'post_id' => '', // Default value.
    ), $atts );

    $output="https://wordpress.stackexchange.com/questions/291524/[acf field="image" post_id="" . $atts['post_id'] . '"]';
    $output = do_shortcode( $output );
    $output="<img src="" . $output . '" />';
    return $output;
}

add_shortcode('my_link', 'my_shortcode');

Usage:

[my_link post_id="xxx"]

where xxx is id of the requered post.