The human ear (for example) is sensitive to frequencies from 20Hz to 20kHz. Any sound beyond that limit would not be heard by us naturally.
Will it harm us (effect our ability to hear) when we hear sound more than 20kHz continuously?
Or...
Will it harm when we hear a 19,999Hz sound (which is upper limit) continuously?
The only sensible answer to these questions is "sometimes".
The reason: even a simple monotone must be described by both frequency and amplitude. Any frequency can cause harm with enough amplitude, and any frequency can be harmless when the amplitude is low enough. If you want to actually quantify the damage, you'll need something like an equal-loudness curve, but for damage rather than loudness. As a proxy, you could use the pain threshold of the equal-loudness curve.
As for the second question, there is nothing magical about 20kHz: physical damage will always fall on a spectrum, there is no single frequency and/or amplitude at which damage begins. If you set an oscillator to constant amplitude and slowly turn up the frequency, you'll notice that the sound slowly dies out around 20kHz. The damage threshold will behave the same way.
If you are talking about sound damaging the sound sensing organs in the ear, analogous to an ultrasonic heavy metal concert, I've found an interesting report just on this topic.
For ultrasonic components above 20 kHz, the limits were set to avoid hearing damage in the audible (lower) frequencies. One-third-octave band levels of 105-115 dB were observed to produce no temporary hearing loss, and were therefore judged non-hazardous in respect of permanent hearing damage.
If you are talking about high frequency sound from a explosion, or some sort of sonic cutter with intensities of such a level of compression or a high decibel range that conveys enough energy to physically damage something of course the answer would be 'yes this would damage your ears and the rest of the body too.
Operators of ultrasonic equipment with levels above 60 dB complained of headaches and fatigue, even nausea below 60 dB they felt no effect (reference table 7 in paper). But they did not seem to have any hearing loss.
One soviet experiment is both informative and maybe also entertaining to read about:
An unspecified number of subjects were exposed for an hour to a tone of 20 kHz at 110 dB. Tests were made to examine shift of hearing threshold over the frequency range 250 Hz to 10 kHz. Pulse rate, body temperature and skin temperature were also monitored. These tests showed no appreciable effect, even when the Sound Pressure Level was increased to 115 dB. These same subjects were given a one hour exposure to a 5 kHz tone at 90 dB: a considerable TTS was found. The 5 kHz tone at 110 dB produced a powerful vascular response.
...
It seems safe to infer an underlying concept: A sound which does not produce temporary dullness of hearing cannot produce a permanent noise-induced hearing loss.
