My dad has dementia, eyes are closed more and sleeps more. He continues to have falls. Are these signs that the dementia is advancing?

Yes, those could be symptoms of his dementia advancing. How much is he sleeping now? All at night, or does he sleep in the daytime, too? Is he fully alert when he is awake?

When do the falls happen? How often? Does he use a walker?

I suggest observing and making notes for several days, and then discussing this with his doctor. Is there an online patient site where you can post such information for his doctor?