According to my teacher, an electron is point sized and it does not absorb or release energy. Moreover, my teacher says their orbital absorbs energy rather than the electron. If that is the case, then what about the photoelectric effect, in which electrons release energy after excitement?
... an electron is point sized
Although it's commonly said that fundamental particles are point particles you need to be clear what this means. To measure the size of the particle to within some experimental error d requires the use of a probe with a wavelength of λ=d or less i.e. with an energy of greater than around hc/λ. When we say particles are pointlike we mean that no matter how high the energy of your probe, or how small its wavelength, you will never measure a particle radius greater than your experimental limit d. That is the particle will always appear pointlike no matter how precise your experiment is.
But this does not mean that the particles are actually zero dimensional, infinite density, dots whizzing around.
and it does not absorb or release energy
It does absorb and release energy. For example, with a laser it is possible to brake electrons which are moving towards the laser source. The electron slows down and lose kinetic energy, it release energy in the form of emitted photons. After standstill of the electron the laser will accelerate the electron away from the laser source. To hold the sum of all involved energy components during positive acceleration an accelerated electron has to absorb some part of the EM radiation from the laser.
... orbital absorbs energy rather than the electron.
This is negotiable and depends from does one think about electrons as particles or as a disturbance of some field. From the particle view it is without doubt, electrons moving nearer to the nucleus are releasing energy in the form of photons. Getting disturbed by a photon of needed minimum energy the electron absorbs partially the photons energy and is moving away from the nucleus. in the case of a field view the electron orbitals absorb and emit the energy.
To answer this question, you'd have to agree on what model of the electron you're talking about. Quantum mechanical? Classical?
Electrons can have force exerted on them by electric fields. If this causes the electron to move, then work is done to it. Thus, energy is transfered "to" the electron.