If you have a closed system (no cap on the radiator, fill cap is screw type on the tank) Yes it will cause an overheat.
It is possible to overheat due to a broken recovery bottle in a normal system but usually it takes a long time as very little coolant is lost and generally it can run with nothing in the bottle for a month or more without problems. In any case you can obtain one for either system very cheap at any local scrapyard.
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