I will answer this, as the chosen answer is not doing so.
If I point two lasers that emits different frequencies with the same intensity at a single spot at the same time, what would be observed?
One has to keep clearly in mind that classical electromagnetic beams emerge from zillions of photons ( which are quantum mechanical entities) in a straightforward but mathematically complicated way.
In quantum mechanics everything is explained by wavefunctions which are the solutions of the corresponding to the particle quantum equations. For photons it is a quantized form of Maxwell equations. . Wavefunctions can be superposed, and the complex conjugate square of the superposed functions which gives the probabilities of detecting the photons, will show interference.
Lasing is a quantum mechanical phenomenon.
This is a very instructive video that shows the interference between laser beams, yes there is interference. The whole system, laser source and setup is in a quantum mechanical state, the power source taking part in the appearance and disappearance of the laser beams.
One has to keep clear in mind that superposition is not interaction in quantum mechanics. The two photon beams are not interacting , the whole system is superposing the photon wavefunctions in specific (x,y,z,t) so that when measured the interference pattern appears, similar to the double slit single photon patterns.
The above video uses the same frequency because interference effects are much more prominent for same frequency waves. But this answer here is relevant for differing frequencies , where for waves in general there will be interference patterns depending on the frequencies. For details have a look at this link. Keep in mind that at the photon level the mathematics of the sine and cosine functions are the same but they refer to probability distributions, not the photons themselves.
Edit: More specifically what happens to the photons when they intersect?
There exist higher order diagrams where photon-photon interactions are non zero. This is shown here,
where the solid line represent virtual particles of the whole gamut of charged elementary particles, the larger their mass the more depressed the diagrams, so usually only the electron loop is shown.
If the energy of the interaction of the two photons in the center of mass is below the pair production energy threshold the photons scatter elastically and get out of the beam line.
The frequency of the photons has to be high for pair production to materialize.
After the electron pair production energy threshold, at least 0.5 MeV per photon, particles may be produced at the intersection of two laser beams. There exist proposals for a gamma gamma colliders.
Photon beams can
be made so energetic
and so intense that when
brought into collision with each other
they can produce copious amounts
of elementary particles.
Energetic refers to frequency, E=h*nu. Intense to the number of photons per cm^2.