Eh, no, because Charles is, no?
I rather thought it self-evident that I was referring to the slot following Prince Charles in the line of succession. Very well, I'll spell it out, then: If Prince William denounces the monarchy and gives up his place in the line of succession, then Prince Harry moves up to follow Prince Charles.
Only temporarily. As Blitzhund says, there would be a media circus for a time, but it would die down and then life would continue as normal. If the British monarchy can survive a king who is a Nazi sympathizer and thinks little enough of his station that he'll abdicate to marry an American divorcee, then surely it can survive losing one king who is fed up with the monarchy and gaining another who only thinks it's funny to dress up as a Nazi at a fancy dress ball. Many Britons would be displeased, but it's not like they haven't endured worse disappointments from the royal family before without everyone suddenly turning republican.