If two sound waves that are different frequencies create beats that occur several hundred times per second, can you hear this effect as its own tone?

No, one cannot hear the actual beat frequency. For example, if both waves are ultrasonic and the difference in frequency is 440 Hz, you won't hear the A (unless some severe nonlinearities would come into play; edit: such nonlinear effects are at least 60 dB lower in sound pressure level).

When two ultrasonic waves are close in frequency, the amplitude goes up and down with the beat frequency. A microphone can show this on an oscilloscope. But the human ear does not hear the ultrasonic frequency. It is just silence varying in amplitude :)

(I know a physics textbook where this is wrong.)

Edit: in some cases the mind can perceive the pitch of a "missing fundamental". For example, when sine waves of 880 and 1320 Hz are played, the mind may perceive a tone of pitch A. This is a psychoacoustic phenomenon, exploited for example in the auditory illusion of an Escher's staircase.

Yes - American Technology Corporation, Woody Norris invented a phased array consisting of ultrasonic transducers; pairs that transmit two ultrasonic frequencies that are slightly different by a modulated sound frequency.

Demodulation of the audible signals from the ultrasonic carriers is accomplished either by nonlinear properties of air or by the two signals striking a surface such as a wall or the inside of your head! In any event the sound appears to occur virtually out of thin air.

These devices have been called hypersonic sound speakers or audio spotlights. Once in awhile you can find them for sale on EBay

As always for anything involving biology, the answer is actually more complicated.

It is true that there is no "note" there at the beat frequency, in terms of Fourier series. But despite what is commonly stated in textbooks, the ear does not just do a Fourier transform.

In fact, the human ear does perceive differences in frequencies, and more generally certain linear combinations of frequencies, as actual tones. They are called combination tones, and a demo is here. As you can hear in the second clip, when two frequencies $f_1 < f_2$ are played, one hears tones at frequencies $f_2 - f_1$ (the difference tone) and at $2f_1 - f_2$ (the cubic difference tone), as well as some others. This is no small effect; these tones are several octaves below the original tones.

This would be impossible if the ear were a simple linear system, because there is no Fourier component at frequency $f_2 - f_1$ or $2f_1 - f_2$. But the ear is nonlinear, and its output is then subsequently processed by the brain, again in a nonlinear way. And it's well-known that the simplest thing nonlinearity can do is output linear combinations of the input tones; that is one of the cornerstones of nonlinear optics.

While the theory is not completely understood, almost everybody can hear the difference tones are there. However, in the case of extreme ultrasound, it's quite unlikely that you'd hear anything because an ultrasound wave can barely budge anything in your ear at all. If your ears are not sensitive enough to detect them in the first place, it's unlikely they would be able to output nonlinear combinations of them no matter how nonlinearly they process the sound.