github.com/rogerjohansson/html5validator (see Downloads for an xpi)
You should not use the validator.nu online service with it, since it causes a DoS attack on the online service, and you will probably be banned from using the service as a countermeasure (see github.com/rogerjohansson/html5validator/issues/6). The cool thing is, it does not send the URL to the validator, but the HTML data directly; this means that local sites or password-protected sites can also be checked.
Validation can be turned on automatically, by a domain-whitelist or by clicking the validator item.
You can and should up your own validator.nu-instance (at least on Linux and Mac OS X) - see this. You would then have a locally-running validator and can even check sites without an Internet connection by filling in http://localhost:8888/ (the default address of the local validator instance) into the Validator URL of the addon.
To give credits: I have the information above from Validating HTML5 with validator.nu and the HTML5Validator Extension for Firefox."
