Briefly -
Cheney's theory is that the lower-than-normal oxygen levels in people's bodies protects them from oxidative stress. The less oxygen we are using, the less oxidative stress is produced.
This is true in general, but of course having less oxygen in our bodies has a downside too!
For whatever reasons, CFS patients have a greater than normal amount of oxidative stress pressure on their systems. This seems related to toxicity. Some of these toxins may be internally produced, but toxins in the environment do that too.
Certain chemicals produce more oxidative stress than others. Toxic mold, such as satratoxin, produces an especially large amount of oxidative stress.
Insofar as people's bodies are functioning with low levels of oxygen, they have some protection against these toxins. If you force oxygen into the system, the toxins become more damaging.
If indeed a bad environment is pushing people into oxidative stress, then increasing the oxygen in that environment is not necessarily a good idea since it just exacerbates the problem. On the other hand, perhaps if people can get away from that oxidative stress, they may benefit from it.
Of course, as with everything in CFS, there are complications. For instance, Lyme infections cause lots of oxidative stress too (Lyme makes a chemical similar to the one made by toxic mold), and oxygen can be helpful in killing Lyme (since Lyme thrives in an anaerobic environment). So for some people with Lyme as their main problem, HBOT may keep the Lyme in check enough to be beneficial overall, regardless of the environment.
And of course, working at a low level of oxygen is problematic because it causes brain dysfunction.
But in general, I think Cheney may be right in that oxygen is problematic for CFS sufferers. Most of them are super-sick and (from what I can tell) living in very bad places, so it seems that he sees that very frequently. Those CFS practitioners who have seen a different patient base (such as Lyme patients) may have had different experiences with HBOT, and I wouldn't necessarily assume that those transfer over to this patient base.
Very little of what I've written is commonly accepted -- it's based on piecing together various literature/theory and on my observations. So take it with that in mind.
Best, Lisa