If we have the connection shown below (star connection of the secondary part of a TX), the load per each phase is equivalent (balanced load), then the vector sum of I1, I2 and I3 is zero. In this case, I would like to ask if we can remove the neutral wire from the connection since no current will flow in it (theoretically at least)?
And if we can remove it, where is the return path (to form a closed circuit) for each phase current?
Since you have a 3-phase system, you can remove the neutral connector if the loads are actually going to remain same. In practice the loads often aren't going to be completely balanced and there will be some current going through the neutral conductor.
Do note that in traditional 3-phase systems, the phase difference between phases is 120 degrees. That in practice means that you'll always have at least one voltage source that will have negative voltage and serve as the return path for the system. Take a look at this simulation. The yellow squares represent the current and as you can see the return path is a voltage source and the voltage sources are short-circuited together on their other end so the return current goes from one voltage source to another.
Here is the simulation of what you posted and you said, no current goes through the neutral conductor. Do note that the simulation does show voltage at the conductor, but the voltage is with respect to ground and not with respect to the center point of the 3-phase voltage source.
Only if all three phases go to windings of the same motor would it be safe to assume that they will stay in balance. In that scenario, if all three phases are supplied balanced power, the current in the neutral wire will be zero, and the voltage would be zero even if the wire were disconnected. If the supply becomes unbalanced, however, the neutral wire may try to play a more active role, for better or for worse.
There is at least one case where the neutral wire should not be connected to the supply. In some installations, three-phase motors are required but three-phase power is not available. It is possible to start a three-phase motor by connecting two of the legs to 240VAC, and connecting the third to a capacitor box. A motor driven that way will not be able to supply any significant torque until it up to speed, but once it is spinning, the third leg will generate voltage which is in the proper phase, relative to the other two, to drive a three-phase motor. One shouldn't try to draw too much current from this third phase, but it may be used to start a larger motor (again, one shouldn't demand torque from that motor until it is up to speed). Note that in such arrangements, the neutral wire of the assembly will not stay at ground potential, and must be allowed to safely "float" or else bad things will happen.
