This happens when you are in a window that is the same resolution as your desktop, You can try one of the many borderless window programs out there (I can never get it to align properly so the mouse is always slightly off and the lines still persist).
Or here's what I did to give me a nicely sized window.
edit the eqclient.ini in your eq folder and change these lines:
WindowedWidth=1910
WindowedHeight=995
Which looks like this at 1920x1080:
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