Need guidance for building a electronic load using BJTs
That's expected base current.  When the switch is open, the feedback loop is disconnected so the 10x amplifier is putting 0V or close to 0V to the comparator, which is driving the 2N3904 as hard as it can to try to reach the setpoint.  The 2N3904 is letting base current from the MJ2955s through which comes from your source.
With this configuration, the base current through the MJ2955s cannot be nulled out since base current is, so to speak, what causes current to flow through them and is therefore unavoidable.  You could switch to an NPN transistor on the low side of your load and shunt resistor so that the base current can be provided by a different source.  You'd probably have an unacceptably high common mode voltage across whatever amplifier is connected to the shunt resistor so a Hall-effect or flux gate current sensor might make sense.