I suspect your problem lies in "zoekterm." Can you give us the values you typing into those two fields? Try taking the exact strings you are typing into the fields and hard coding the SQL query. Do you get the same error?
Your WHERE statement should say WHERE thisfield LIKE 'something' or WHERE thisfield = 'something.' If your zoekterm contains double quotes or no quotes at all, then it's probably causing your error.
Another thing to try when creating your SQL code is this:
$SQLString = "SELECT * FROM " . $POST['zoekveld'] . " WHERE " . $POST['zoekterm'];
$PostcodeLijst = mysql_query($SQLString);
This way, you can check the value of SQLString by echoing it to the screen, so you can check to make sure it's being created correctly.
Hope that helps.