Can beats be produced by two waves moving in opposite direction?

Two waves moving in opposite directions that have the same amplitude and frequency produce a standing wave. If one of those waves has a slightly different frequency than the other, it should produce a "standing wave" that slowly moves. As the anti-nodes and nodes move past an observer's ear, they would hear an increase in amplitude followed by a decrease in amplitude, and this variation in sound amplitude would be interpreted as beats.

Generating such an effect would be "tricky". A sound generator would have to broadcast a pure tone and face a sound reflector such that a standing wave was created. Then, the sound reflector would have to be put in motion, moving either directly towards the sound generator or directly away from it such that the Doppler effect slightly shifted the frequency of the reflected sound. If the sound reflector was very efficient (close to 100% reflectivity) and a small microphone was placed in the path of the reflected sound waves, it should be possible to record the effect.

Beats are produced by the superposition of two waves at a specific location. The superposition is the sum of the displacements produced by the waves, not by the direction of velocities of the waves.

If you're considering longitudinal (sound) waves, there are compressions and rarefactions produced along the paths of the waves. If we consider sinusoidal waves, the compressions are $lambda/2$ and $lambda '/2$ apart and travel at speed $flambda= f'lambda '$. The superposition occurs at a point in space. The beat frequency is a consequence of the compressions matching or cancelling each other at that point. The direction from which they arrive has no bearing on how often they reinforce or cancel.

Hearing beats is actually a psychoacoustic phenomenon. If the frequencies are different enough you will hear two frequencies rather than one, beating. The point at which that occurs varies from person to person. The mechanical effects of beats depend on the coupling of the structures experiencing the two source frequencies. It may be benign if the beat or the average (higher) is not a resonant frequency of the structure, or the beat could be destructive if it is near a resonance and not properly damped.

Again, the velocity directions of the waves are unimportant for the beating effect to occur. For acoustic beating, consider that the sounds are bouncing off the walls and ceiling and floors, so the waves are coming from all directions.

This answer expands on the anwer by David White.

Take the case of two sources of propagating waves, let's make that sound sources.

Sound spreads in all directions, so the larger your distance to the source the smaller the amplitude of the sound that reaches your ears.

When the two sources are right next to each other then wherever you are positioned the two waves will have pretty much the same distance attenuation. So you can move closer to the source, or away from it, the beat frequency phenomenon remains roughly the same.

Now the case of the same two sources, but now they are a signicant distance apart, and you are positioned somewhere in between.

The two sources emit sound with a slightly different frequency, so there will be a beat frequency. Except, as indicated by David White, there will be only a single "sweet spot" where your distance to the two sources is such that the amplitudes of the sounds as they reach your ears are roughly the same.

Everywhere else you still get some degree of beating, but not as pronounced as when you are standing at the "sweet spot".

Overview:
The only necessary condition for beat frequency is that you have two sources that are off by a small fraction of their own frequency. Alignment of direction of propagation is immaterial.

It's just that if the two sources are not right next to each other the full wave cancellation will occur only at a relatively small "sweet spot".