How do I remove the shadow cast by the rectangle using the cycles renderer?
I do not think it's a shadow, but more of a spot the emission of the plane cannot reach because it's not pointed to that direction. Would giving the plane a bit of depth solve your issue?
Edit: Another solution i could think of is replacing the plane with a thin cylinder.
Extrude the plane slightly or use a cylinder. What you see is not the shadow, rather the geometry emits no light to the ground in this spot.
In case there are multiple light sources in the scene, disable the Shadow property for each emitter (Object Properties > Cycles Settings), see How to render an object without its shadow?
Try this node set up The lightpath acts a fac into the mix so the shadow rays are transparent.