The answer to your question is yes, we can observe beat notes between two different coherent sources of light. This fact underlies almost every precision laser experiment because it allows for lock-in detection. However, there is a subtle difference from audio beatnotes.
The difference is that with sound the oscillations are in the air pressure which is what you detect with your ear. The beatnote you get with two sound waves is in the volume of the field. Consider two sound waves with unit pressure combined at your ear,
$$
P=\cos(\omega_1 t)+\cos(\omega_2 t)=\boxed{2\cos\left(\frac12(\omega_1-\omega_2)t\right)
\cos\left(\frac12(\omega_1+\omega_2)t\right)}.
$$
The first term is the beat note that you detect. It is a beat note in the volume of the strength of the signal. However, if you were to take the Fourier transform of this signal you would still find that there are only two frequencies: $\omega_1$ and $\omega_2$.
With light, the frequencies are so high, that all we can detect is the intensity of the field. This is an inherently non-linear process in which the beatnote actually becomes a component of the signal. Consider two optical fields of unit power (using complex notation this time),
$$
E=e^{-i\omega_1 t}+e^{-i\omega_2t}.
$$
When this field is incident upon a photodetector, the signal will be given by the time averaged intensity of the field
$$
\langle E\rangle=E^*E=\boxed{2\left(1+\cos[(\omega_1-\omega_2)t]\right)}.
$$
So, this signal actually has a component at the beatnote frequency. I.E. if you take the Fourier transform you will find a component at that frequency.
Why does the difference matter? With light you can combine two lasers whose frequencies are much too high to be detected directly and obtain a signal at the beatnote frequency. You can not, however, combine two sound fields with frequencies which are outside of the range of human hearing (above $\sim25\ \text{kHz}\ $) and be able to hear the beatnote between the two.