The security error is unlikely to break your page. It looks like the error is happening from within the YouTube frame, which means that in the worst case the content of the frame will be messed up.
A frame/iframe from an external page cannot, under any circumstances effect the content of the parent document unless they are from the same domain and port number. It is one of the hard rules of browser security.
The error must be elsewhere in your markup. Any chance of seeing some example markup?
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The error could also be in the embed code markup. Or, if any script tags are included directly on the page (not in the iframe) it could be there.
Usually when problems like this happens it is because of an unclosed tag somewhere but it could be Javascript as well.
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