After clicking on Add Input and adding an input, clicking on the added button doesn't call the required_valid() function in $('.submit').submit(function () {.... Why is this?
DEMO: http://jsfiddle.net/Jvw6N/
<a href="" class="add_input">Add Input</a>
<div class="get_input"></div>
$('.add_input').live('click', function(e){
    e.preventDefault();
    $('.get_input').append('<form class="submit"><input name="test" class="required"> <br /> <button>Click Me</button></form>')
})
function required_valid() {
    var result = true;
    $('.required').each(function () {
        if (!$(this).val()) {
            $(this).css("background", "#ffc4c4");
            result = false;
        }
        $(this).keyup(function () {
            $(this).css("background", "#FFFFEC");
        })
    });
    return result;
}
$('.submit').submit(function () {
    var passed = true;
    //passed = required_selectbox() && passed;
    passed = required_valid() && passed;
    if (!passed) {
        $('#loadingDiv, #overlay').hide();
        return false;
    }
});