Speaking about tube amps with an OT, and not SS with no OT.
Actually, too high an impedance speaker, up to and including an open, can damage an OT, while a lower impedance speaker is fine. Example, using the amp with no speaker or no shorting jack attached to the output can damage the OT. At some point the impedance becomes too high, but a 2 to 1 mismatch either way will be fine. An 8 ohm amp can easily and safely run a 4 or 16 ohm speaker. Even a 2 ohm speaker on an 8 ohm amp would be fine, but a 32 ohm might be too high.
Using a lower impedance speaker, down to and including a short circuit across the output is fine, no worry of damage. Many amps even have a shorting jack that will short the OT secondary if the speaker is unplugged, and this will protect the amp from damage. If a short circuit on the output protects the amp, then a low impedance speaker load won't do any harm either.
Sometimes a mismatch can give a better sound than matched. The best sound from my Twkr is with the amp set to 16 ohms and using an 8 ohm spkr cab. The sound is just a little crisper than with the amp set to 8 ohms. A 4 ohm amp like a Champion 600, sounds better to me with an 8 ohm speaker than it does with a 4 ohm speaker. Using the entire secondary of the OT (highest impedance setting) will give the best sound with various impedance speakers IMO.
Remember that the speaker impedance is a nominal value determined at one specific frequency (typically 1KHz), as the actual impedance of a speaker at any point is determined by the frequency going to it at that particular point in time. If a 2 to 1 mismatch caused any harm to amps, then every amp would blow up unless you only played a 1Khz tone through it at all times.
