Is there any way to make Gmail automatically archive an existing, un-archived thread when a new message on that thread arrives that meets certain conditions?

is correct. To add value, I'll offer a workaround for the desired behavior you described in the question details (i.e., not requiring users to archive or mark as read notifications emails for notifications they've already seen/read on your website). You could delay the notification emails by a period of time (fixed or variable), so that it gives users a chance to see the notification on Quora before the email is sent. As such, users who are active on the site won't be bombarded by the emails. In addition, the period of time could be variable to be more scalable and effective for very active as well as less active users. This would be accomplished by making the notification email delay a function of the user's activity on the site; for example, if I am active on the site every 2 hours, on average, Quora would wait 2 hours before sending me the notification email. Conversely, if I am active on the site every 2 days, on average, Quora would wait 2 days before sending me the notification email. The result of this dynamic email delay would be that users would less often get notification emails when they had already seen or read the notifications on Quora itself. In addition, Quora would still be successfully alerting users of notifications they haven't seen and pulling them back into the site. (For what it's worth, I discussed a variation of this implementation in )

Open the email that you want to apply the action to.Click on the "More" drop-down-menu and select "Filter messages like this".Set up the conditions (from:quora or something like this) and then create a filtering rule.There is an option "Skip inbox (archive)".Also Upsafe backup tool allows you to set up automatic Gmail archiving. Check the article below:http://www.upsafe.com/how-to-archive-all-gmail/

I wrote a Google App Script that will archive messages based on a time period criteria.  While this module is not exactly what you are looking for, it is a custom auto archive script for Gmail.  The script can be modified to serve your purpose.  So in short the answer is Google App Script and the "almost" proof of concept is the little module I created (see link below). https://script.google.com/macros/s/AKfycbwrmQoIdg-_TQmzVQC9rAVgJu0ZM5UbJ4r06TlsBTPcgAq43aA/exec The tool can be scheduled to run automatically via what Google App Script calls "triggers".

I disagree, although I may get some slack on technicalities. There is (or was) a Gmail lab option for "MUTE."  By muting a thread it will automatically be archived.  It may not entirely accomplish the goal that you want however, so I do recommend trying it with a few test cases. Mute- functions as a label with a built in filter. I'm on several group lists and periodically a thread gets out of hand and I've used this option.