I have occasionally noticed a series of beeps at the beginning of a
commercially produced VHS tape. The beeps sound a bit like someone
dialing a phone from a distance, and I seem to remember that they vary
in either frequency or volume. What is the purpose of these sounds?

Your question creeped me out.  I am a sucker for conspiracy theories,
subliminal messaging, government mind control schemes, and mysterious
multi-tone sequences that activate directives implanted in our
subconscious by the alien master race.  I cannot wait for someone to
answer this!

This "source" doesn't really say why/how they know, but they do claim
to have the answer:
http://www.answerbag.com/q_view.php/7907

These beeps are actually the first bits of encoding that keeps you
from copying the tape. I know this because I am a movie collector who
recently took all my VHS tapes and converted them to DVD. When doing
this I noticed that my newer tapes would not record. My relative who
owns an online movie/toy store informed me what the problem was, and
gave me a "Little Black Box" as he called it, that overrode the
encoding and allowed me to record my tapes. No government control, or
anything like that is taking place, there are no conspiracies. Just
big movie companies covering their rears to keep people from making
illegal copies and keep their pockets full.

BTW, my relative runs a movie store and copies movies that the
American copyright has run out on, and has even bought the rights to a
few movies. His operation is completely legit, if you'd like to check
out his collection the link is www.monsterlandtoys.com