Electronic – Does a TIP120 transistor have an internal resistance when switched on

One, that switch does not directly control the motor. It's most likely a few mA at best, as it signals a microcontroller inside the cdrom to open/close the tray.

Two, what you are looking for is simple ohms law. Resistor = (Source Voltage - The Transistor Base/Emitter Drop) / Current Required in Amps. Since the hFE or current multiplication ability of the TIP120 is 1000, so roughly it will allow 1000 times the base current at the collector, any amount of current should be good at the base. Let's just use 5mA. The Base/Emitter drop is 1.25V minimum, as there are two transistor diodes.

Resistor = (5v - 1.25V) / 0.005A or 3.75V / 0.005A = 750Ω or close.

Update To further answer the question, you calculate the base resistor within the safe range of your source (Arduino, 40mA per pin, 200mA total at any given time). The unknown collector current in this case would be minimal for a button. For actual loads like a motor, you could simply max out the transistor by saturating it, giving the base transistor as much current as possible. In this case, you would have to have multiple arduino pins in parallel since the TIP120 base limit is 120mA and the Arduino is 40mA per pin. This is not ideal because you don't know the current at C-E or the amount of voltage it will drop.

The best answer is that you DONT. A proper design will find out how much current will be at the collector. Use a ammeter or multimeter in current mode to find out.

The main purpose of the base resistor is to limit excessive current to the base. The BJT is a voltage controlled device and hence current is not the driving factor for switching. The base resistor needs to be large enough to prevent damage to the transistor, but should still allow sufficient current to ensure the transistor switches on and off as per the base voltages.

Not using a base resistor sometimes works, but it's a terrible practice and it's just asking for trouble. Relying on this mechanism runs the risk of burning out your I/O pin as well as damaging your transistor, so its recommended you use a base resistor. Without a resistor, you are placing 5V on a low impedance input (Base - Emitter), and asking the Atmega/Arduino pins to source a lot of current. Eventually, if not immediately, that is going to destroy your Atmega.

However, you do not need a base resistor if you operate the transistor in the common collector configuration sometimes called an emitter follower. This is because any current flowing through the emitter load cause the voltage on the emitter to rise to a point where it is 0.7V below the voltage on the base and prevent any further current flowing. This is a sort of negative feedback on the base current.