Most people add NYT pages from the “Read Later” bookmarklet or the new in-app browser. Pages added from both of these sources are unaffacted by paywalls and login requirements since they both save copies of the page as the viewer sees it.
In other words, if you can see it, Instapaper can save it for you with the bookmarklet or in-app browser. So you’ll still need to pay for a NYT membership if you exceed their monthly thresholds for viewing.