We are told that elementary particles has dimension zero and take up no space. For example, the electron is a point particle that have a negative unit charge, also has mass and spin, but no size.
My question is what bad consequence does it lead if we assume that electron has size? Is there any direct or indirect experiments that verify this assumption (or facts?), i.e. elementary particles have no size. Or what theoretical contradiction will it have if we do not assume point particle?
If the electron is supposed to have a finite size, then it has has to be a rigid body. If not, then it wouldn't be an elementary particle, it would have some inner structure that you can further investigate. So, if the electron is indeed an elementary particle and it has a finite size, then it must be a rigid body. But a rigid body can be used to create causality violations. Information can be transmitted instantaneously across the distance covered by the rigid body (if a force is applied to one side the other side will move instantly). As explained here in detail you can exploit that to send information back into your own past.
This is why in physics we assume that there are only local interactions. An effect at some point in space at some time can only have an effect at exactly the same point at the same time. Rigid bodies cannot exist, forces are transferred from one point to another via fields that only interact with each other in a local way (the interaction terms only involve the field strengths at the same points).
At school, and when studying classical physics, one assumes that particles have no size. This is done because it massively simplifies equations, and it is a good approximation if the size of the particle is believed to be significantly small than the other important length scales involved.
Actually, there are unwanted consequences when one assumes that a particle has no radius. For example, the electric field of an electron increases as $1/r^2$ as one gets closer to it. Since the electric field energy is proportional to the square of the electric field magnitude, the energy of the field of an electron in vacuum would be infinite!
In fact, in modern physics, one does not talk about the size of a given particle. Rather, this concept has been replaced by that of the probability of finding the particle at a certain place when looking for it. This uncertainty in the position of the particle can be used as a measure of its "size".
But this "size" does not behave as our normal intuition of size. For example it is variable, and generally depends on the momentum of the particle in question: the higher the momentum, the smaller the size can be. This is connected to the Heisenberg Uncertainty Principle. This is confirmed experimentally. Slow neutrons in a nuclear reactor are far more likely to collide than fast moving ones. If you want to make sure an electron passes trough a slit, you better fire it with high momentum.
In the same way, one can, by using higher and higher energies, determine the position of an electron to higher and higher precision, thus reducing its "size". But to reduce it to 0 would involve, you guessed it, infinite energy.
