Thanks, everybody. lsmartino and b, your suggestions led me to try
the following:

Toshiba SD-V296KU connected to HDTV, and switching the Toshiba's audio
modes back and forth:
Tape A - Hi-Fi mode: Static
Tape A - Mono mode: No static
Tape B - Hi-Fi mode: Static
Tape B - Mono mode: No static

JVC HRXVC14BU/HR-XVC15SU connected to HDTV, and switching the JVC's
audio modes back and forth:
Tape A - Hi-Fi mode: Static (at beginning of tape, it comes on as soon
as JVC displays "Hi-Fi" on monitor)
Tape A - Normal mode: No static
Tape B - Hi-Fi mode: Static (but not as loud as when played on
Toshiba)
Tape B - Normal mode: No static

The normal/mono modes sound awful in all cases - loud tape hiss, which
is to be expected, I guess.

More tests to follow.