So far as I know, Church law does not expressly forbid it, but that does not make the practice prudent or palatable. When someone agrees to take on the responsibility of the spiritual formation of another, he or she takes on a kind of spiritual parenthood. (That’s why the office is called godparent.) At the very least, romantic involvement may compromise the godparent’s objectivity and freedom to call the godchild to spiritual account. At worst, it could be a form of spiritual abuse because the spiritual authority the godparent has over the godchild can compromise the godchild’s free consent to a romantic relationship.