The HNQ is a hot-button issue but I think it's safe to say that the way the HNQ visitors interact with a site can often be problematic. This question on preventing "drive-by upvotes" from non-regular users seems to indicate some consensus on that at least. Another suggestion was in: Breaking the HNQ feedback loop on bad questions. Since it's clear that HNQ is here to stay, I'd like to suggest a change that I believe would help balance the entertainment value of the HNQ against the negative impact it currently often has on questions that are featured there: open questions followed from a HNQ link in read-only mode.
A purely technical request for such a read-only option has been suggested before but got less than 100 views and little traction. As mentioned on that question Reddit implements a similar system called "No Participation" and I envision this working in much the same way:
Users who follow a link from HNQ to a different site open that question in a non-intrusive read-only mode and have to take some kind of action to disable that mode before they can interact with it (comment, vote or flag as based on the usual reputation requirements). That action could take a number of forms and could for instance have a site-specific text that briefly explains the culture of the site and the standards it has for answers.
I'm not suggesting a particular form of implementation here as there would have to be support for the general idea first, both from meta and presumably from SE. I assume there'd be some UI coding involved but this has the advantage of not touching the reputation/privilege system at all.