May i know what is the format of the images on which you are getting the Lines.
Also export the images out of the Lightroom and then manually upload them to Flickr and check are you still getting the lines.
Thanks for the suggestion. I followed your instructions and created a JPEG manually and then manually imported it via the web loader. The result was no lines on the file or within Flikr.
I guess this suggests the Lightroom plugin is the source of the problem. Is there a way to fix the plugin or get another one that works, do you know?
Currently, i am using the Flikr plugin 126.96.36.1991433 in Lightroom 4.4