I am experiencing the same issue since i've joined Flickr (2 weeks ago) and started right away publishing my photos via the Lightroom publication service to Flickr.
Basically Lightroom correctly find when a photo has been modified in Lightroom after publication to Flickr (it categorizes it under "Modified photos to re-publish" section, however when i publish it will add an additional photo on my Flickr account, with its own views count, as opposed to updating the existing one, and keeping its existing view counts.
Does anyone know how to fix this issue?
For what it's worth, here are extra information on how i am working:
-using Lightroom 6 v6.61 (latest one)
-on a Macbook Pro running OS X El Capitan (v10.11.6)
-i have often marked the photos as "marked as up to date" so that Lightroom doesn't offer to re-publish them while i wasn't done with all my changes - i presume this shouldn't be an issue, but who knows.
thanks in advance for any help, this re-publish feature was a key asset of the Lightroom publication service in my view, but for me it just... doesn't work :-(