Completely Remove Year From Birthdate

Ummm, Your best chance is possibly hide the year component with css and then use the filter to modify the way the birthdate is displayed if you want.

You need also to fix the year to a leap-year to be sure you have 29 days in february just in case somebody needs it. You can probably achieve this with javascript or using the filter: “bp_get_the_profile_field_birthdate” you can change the selected year value.

add_filter( 'bp_get_the_profile_field_birthdate', 'my_profile_field_birthdate', 10, 2);
function my_profile_field_birthdate($html, $type) {
    // Check if $type is "year" and not "day" or "month".
    if ($type === 'year') {
        // Modify to only an option with a leap-year like 2016.
        $html = '<option value="2016">2016</option>';
    }

    return $html;
}