Should you let the editor of a paper under review know when you post preprints of additional papers that build on the paper under review? Let's say that one author has a paper under review, and continues on subsequent works based on the paper under review. 

When the author puts preprints of subsequent works available online, should the author let the editor of the first paper know about those subsequent works?
No, there is no need to do this.

If you submit a work to a journal you submit it as is and related to the current state of the art at time of submission. Things that happen after submission should in general not interfere with the current submission. While it could be in some exceptional cases that events after submission may influence the perception of the article, in general the article should be processed with respect to the date of submission.

If I imagine that people would start sending any kind of "additional information" after an article has been submitted, I would expect that this could lead to a great mess (articles submitted which are in fact only preliminary, additional appendices added at a later stage, corrections/amendments…). So I would guess that an editor would generally ignore the additional information and even may find this a bit strange.

However, you may add additional information in case of a revision/resubmission and this is the right time to add anything that could be relevant (with the only exception of severe error in your work - if you find one inform the editor immediately).