No, it will not. The thermotime switch only receives power from the same circuit that causes the starter to crank over. It has no power other than that. The only thing that can really cause the cold start injector to stay open is if it physically sticks. If you disconnect the connector to the thermotime or to the cold start injector, the cold start injector should cease to function. If you want to find out if the cold start injector is stuck open or causing your rich condition, you need to disconnect the fuel line to it and plug it. This would also be a good time to 'search' the archives for rich running conditions.
I would check out the water temperature sensor. Does your car run "normal" when it is cold and then starts to run rich as it warms up ? If the answer is yes, I would say you water temperature sensor is busted or has bad connections. One easy way to test if this is the problem is to let the car warm up and then disconnect the sensor. When you disconnect the sensor in a good working car you will hear a difference in the idle and the car will run rich (this is because the computer thinks your car is cold and therefore lets more gas through the injectors). The water temperature sensor is the one above and to the right of the Thermotime switch you can disconnect it at the sensor or at the bullet connections about 12 inches from the sensor. I would go with the bullet connections.
P.S. Another possibility is a clogged return fuel line. This would cause too much fuel pressure and too much gas through the injectors.