No. Ensuring that CARES did not adversely affect the functionality of the tray was an important FAA criterion for certifying CARES. The tray is lowered only while the red webbing loop is adjusted around the seat, and then is closed and locked in place, covering up the CARES loop. CARES does not affect the functioning or use of the tray. Using the tray doesn’t affect the safety or functioning of the CARES restraint.
In some older style seats found occasionally on small planes, the tray fits into a plastic cavity with recessed sides. In these the CARES red loop, when pulled taut, could prevent the tray from closing securely. In this case, the CARES loop should be installed around the seat and OVER the stowed tray for taxiing, take off, and landing. (See FAA Advisory Circular 120-87B “Use of Child Restraints on Aircraft”, which can be found in the “In-flight Documents” section of this website http://kidsflysafe.com/instructions/in-flight-documents/)