May be this will help. You haven't used df and dt parameters in SQL query:
$result = mysqli_query($con, "SELECT trallevask.regnr, trallevask.date,
trallevask.type, equipment.eqname AS vaske_type
FROM trallevask JOIN equipment ON equipment.eqid = trallevask.type
WHERE userid = ". $_COOKIE['userid'] . " AND trallevask.date
BETWEEN " . $_POST['df'] . " AND " . $_POST['dt']);