Assuming you don't want to change the markup: The easiest way is to use a library. e.g. with jQuery:
jQuery('.classname').remove();
Otherwise, you need to get a reference to the element and then:
el.parentNode.removeChild(el);
Getting a reference to the element is the tricky part. Some browsers implement getElementsByClassName, but you'll have to write your own or use a third party implementation for the rest.
if (!document.getElementsByClassName) {
document.getElementsByClassName = function (className) {
/* ... */
}
}
If you are willing to change the markup, then give the element an id and use document.getElementById to get the reference to it.