I tried to put the HTML5 'required' attribute on a textarea that had a wysiwyg on it but the form then became unsubmit-able. The same error occurs for both the tinyMCE and CKEditor.
This is very relevant since there is an ongoing HTML5 initiative to support these attributes on form elements as part of core. See #1174938: Natively support the HTML5 #required FAPI property
I'm not sure if this is something that is an in-built problem with the third-party wysiwygs themselves, or just the way wysiwyg.module integrates them into Drupal. I hope the latter - much easier to fix!
I'd love to supply a patch, but I'd have no idea where to start. Ideally the way wysiwygs work would change so as to update the underlying textarea as-and-when the user makes a change to the wysiwyg, rather than on form submit. This is because there will probably be other html5 attributes that it would be good to support (e.g. 'pattern' - allowing us to supply a regexp for the browser to validate input against) also.