Do piano tuners listen to the beat frequency between their tuning fork and the wire they are tuning?

Yes, they do. When the tuning fork (or the more modern electronic tone generator) is providing a reference tone, the tuner will strike a key and listen for a beat frequency between the reference and the piano string. With wrench in hand, the person tuning the instrument will take a bit of tension off the string, and will then increase the tension to bring the piano string "up" and equal to the frequency of the reference. The beat frequency will disappear as the tones become equal in frequency. It is the practice of the individuals tuning a piano to always bring a string of the instrument "up" to the frequency of the reference rather than "detuning" the string to lower the pitch and match it with the reference. With a bit of practice and patience ('cause you can always detune the string and "start over" to get it spot on), you can generally do a pretty good job of tuning the piano, though the professionals have been doing it for many years. These experienced folks have a good "ear" for the beat frequencies. The electronic references are modestly priced now, thanks to 21st century electronics. Note that there are cool electronic tuning units that will "listen" to the beat frequency and indicate to you when it disappears and a match has occurred. Our ears are generally fairly sensitive to the difference in the frequencies of two tones. When the tones "beat" on one another because they are being generated simultaneously, the difference between them is usually fairly obvious. Oh, and you are listening to the interference frequency between the two tones, which is what the beat frequency is. Certainly it's a bit of a challenge to accurately tune a piano, but many folks are fairly capable of doing it and only need a modicum of practice. Leave that big Steinway or Yamaha to the experts, but if you've got an old upright, have a go!