What is the effect caused by ammeter resistance when an ammeter is inserted into a circuit to measure the current?

The effect the multimeter might have on the circuit when inserted to measure the current is to increase the circuit resistance and decrease the available voltage to the circuit. This is because the multimeter in amps or milliamps mode does have a small resistance which is not zero, so by Ohm's law, there is a voltage drop across the multimeter; small, but not zero.

Usually this effect is small. One way to compensate is to start by measuring voltage, and then inserting a separate ammeter and adjusting the power supply to match the original voltage. Of course, the voltmeter must be downstream of the ammeter.