Are there dangers in being upside down?

Does it strain the heart or brain, for example?
Are there cautions against such taking such a position for extended periods, or if you are over a certain age, for example?

Short answer: yes.

Long answer: your cardiovascular system has evolved along with the body to function best when you’re upright. The reasons for this are numerous, but include skeletal muscle pumps in the legs to make sure blood doesn’t pool there, and baroreceptors in the sinuses that monitor pressure. They would get seriously f-ed up when all that blood pooled there and made your body think your BP is higher than it is.

Bats sleep that way, and it makes them, well, batty. And some say that’s why the Australians are so Australian.

Hey, it’s a joke, okay? I love you folks in Oz.

Just so it’s clear, it should be noted that the ‘sinuses’ you’re referring to have nothing to do with the sinuses in and around your face (i.e. the ones that can get congested or develop sinusitis).

The sinuses involved with monitoring and responding to blood pressure are in the internal carotid artery (the main artery giving blood to the front part of the brain), i.e. carotid sinus baroceptor.

Are you asking about something like yoga, or about hanging upside down for extended periods?

No, not specific, just in general. Lying down is considered healthful when you are ill. Standing up is better when you are drugged and want to keep awake.
When are people upside down? Well, when you have that boot therapy where you hang upside down. Or headstands for fun, not yoga. I used to do them when I was little, but kids are bad examples, since their short, small bodies make a lot of things less stressful, like falls, just from the physics of it. There’s being upside in a rocket during takeoff and reentry. There’s being upside down at amusement parks for various periods depending on the ride.
And of course there’s the torture, perhaps just in cartoons, of prisoners in a dungeon hanging upside down. I wonder if that’s deadly or if your body would adjust in an hour or so and your flush return to normal?