I think I don't exactly follow the problematic here Chris. There's basically two types of logout scenarios. The first one is logging out directly through XenForo. Leaving some information on the device in that case should be fairly trivial, for as long as it can only be used when logging back in with the given user, pausing shouldn't be an issue?
The other scenario is a logout by deleting cookies/browsing data. As XF won't have knowledge about that unless the user visits the page again afterwards, I assume this will just happily continue push notifications anyway?
Alternatively, the user could be asked whether he wants to continue push notifications even though he logs out when starting them. Not sure how to best present that option without making people accidentally activate it though.