My dad, who has dementia and COPD, will start shouting that he is falling while sitting in a chair. Any ideas why this continues to happen?

Such feelings are common in end stage COPD esp if coupled with aging brain and dementia of any sort. This may have to do with brain balance issue, but if you recall you will sometimes get such feelings in bed when near sleep, as tho you have suddenly fallen.
The brain is an absolute marvel but it is the system we know next to nothing about. A neurologist would best answer your question.
This from the internet may help.
Falling Feeling While Being Awake - Neurology - MedHelp

https://www.medhelp.org/posts/Neurology/Falling...
Some describe dizziness as spinning sensation, others lightheadedness or imbalance. The most common cause of dizziness is benign paroxysmal positional vertigo (BPPV). However, you …"
There is much more on the internet so look up "the feeling of falling when awake."

WendyP, same thing happened when my Mom was in the hospital. She kept yelling she was falling, she also was reaching up and moving her legs very quickly.

The nurse told us it was delirium, and once Mom received a shot to calm her down, the delirium became much less.

My Mom didn't remember that happening. It was pretty scary to us.

My mom deals with similar stuff from end-stage COPD. I'm not sure if it is sundowning per se, or just something similar due to lack of oxygen, like climbers of Everest who run out of oxygen then decide to do strip teases. It can come and go based on her activity levels. Right now her pulseox drops like a stone if she tries to stand up too long. At one point she walked around WAY too much and when I got her to finally sit down she started shouting "Where's my husband" and I lost my s**t. (crybaby. im a crybaby.)

She's on hospice here now, and the hospice nurses are saying this is just part of the territory with end-stage COPD at this point. I've been able to keep her oxygenated medded and fed better here and I think some of that has receeded because of that.

When my mom lived alone sometimes she'd call me in a panic because she felt she had been blacked out for over a day. And I'd be like, "Mom you called me a few hours ago".

I will continue to blame lack of oxygen cause I don't like the alternative and in the end as long as the symptoms are kept in such a way that mom is safe and healthy as possible its all good.

People in my family have occasional bouts of vertigo so I'm familiar with the feeling. Do you have a blood pressure machine? A drop in BP is one of the things that would be easy to check.

Wendy, also be aware that a UTI can cause a sudden change in mental status.

Consider buying some at home UTI strips to use. They aren't as reliable as the urine culture a doctor will order, but they can be your "early warning system".

I have not talked with doctor yet, as it just happened yesterday. But I will tell the doctor today. Last night was rough. For some reason he took off all his clothes and was running around naked. Finally fell asleep again and woke up still naked (couldn’t get him to dress in the night) he was wondering why he was naked. I am an only child , we do have a day time caregiver, but he is becoming a handful

This might be an "inner ear" problem. Sort of like when you are starting to fall asleep then suddenly jerk. Sometimes it happens to me when I am sitting on the couch.
Has he had mini strokes before? (probably not worth putting him through testing to find out if that is what is happening.)
You might want to try one of those weighted blankets to sort of "ground" him. I would get one of the ones for kids as it would be more like a lap blanket for him and it would be a lot lighter in weight. (I got one for myself a few years back and returned it, it was so heavy I could not turn or even move under it.)
I would definitely let his doctor know of the incidents.

This sounds like a balance issue.

If you Google "feeling of falling while seated" you will get a lot of information on brain issues that can cause this.

Have you spoken to his doctor?