It seems (and probably only SEEMS) that when I am awake, it requires effort to close my eyes. When I’m tired, it requires effort to open my eyes. Are there muscles working here that change their objective depending on my state of alertness? Are different muscles doing the two different things? Is it something else altogether?
Ahhh…the wonderous human body.
Not quite so wondrous when you’re stuck on the toilet for an hour or trying to get rid of a pimple, but. 
To answer your question: you’re thinking of the levator palpebrae superioris muscle. You can control it voluntarily (shutting your eyes), but when your brain shifts into sleep mode your body shuts down all your senses and forces the eyelid to close involuntarily.
“…shuts down all your senses…”? That doesn’t seem right. I hear something loud, I wake up. It’s never happened, but I hope I might smell something burning and wake up. I might even feel the heat and wake up. I’m doubtful I’ll taste something that would wake me up, but I imagine it could happen. And we’re left with sight.
But do the senses have anything to do with it? I know some blind people keep their eyes open during the day, and I assume they close them at night. It doesn’t seem to have anything to do with our sense of sight.
Even if it were a sense thing, it’s still the only muscle affected. My heart keeps beating while I sleep. My bowels don’t loosen. (OK, they usually don’t loosen. HA!)
Actually, it seems (again with the “seems”) that my mouth generally stays shut while I’m awake and not talking. But I know that I sleep with my mouth open (ick…all that drool). I don’t think I’m thinking about keeping my mouth closed all day. But am I? Is the jaw also affected by wakefullness?
If you don’t wake up when someone turns the light on then you are unusual. For most people sight works just fine while they are asleep.
It’s not the only muscle affected. All your skeletal myuscles are paralysed when you fall asleep. Your heart and bowels work just fine because they are not skeletal muscle and are under the control of the autonomic nervous system. All your conscious muscles undergo flaccid parlysis when you fall asleep, which is why sleeping people go limp. The muscles controlling the eylids are no different.
Yes, and yes.
People maintain a “face” when they are awake. That’s why we are able to read so much about aperosn;s mental and physcial state by their facial expresions. You keep your jaw in certain position, and your neck, and your eyebrows and every other part of your face. You don’t notice that you are thinking about to because you’ve had years of practice, just like you don’t think about walking. Nonetheles both walking and maintaining facial expression are conscious actions that require active thought for maintanence.
And just as people stagger as they get tired so people’s faces droop as they become increasingly tired.
Actually, there’s just a little counterweight at the bottom of the eye. When you’re vertical, the weight tends to hold your eyes open, but when you’re horizontal, the weight pivots the eyes closed so it’s still at the bottom.
I mean, that’s obviously why they make those dolls that way, isn’t it? They wouldn’t do it that way if that weren’t how real human eyes worked.
Don’t count on it. As I understand it, in most house-on-fire situations, you’d be long dead from smoke inhalation and/or CO poisoning before you had a chance to feel the heat.
Your eyelids are under both voluntary and involuntary (autonomic) control although the “voluntary” part is a bit of misnomer; it’s a very automatic response to the ongoing stimuli of the visual world when we’re awake. You are right; it doesn’t seem like you have to think about it. (But then, you don’t think about walking and yet that’s a “voluntary” activity which you can change at will.) For the most part, though, it’s not an autonomic (involuntary) response except for a small contribution from the sympathetic chain that innervates the superior tarsal muscle, contributing a couple millimeters of additional opening. If you lose the function of this muscle, your eyelid will droop a little (mild ptosis). However the voluntary muscle is the levator palpebrae superioris, and this is what mostly keeps your lid open; it’s a voluntary muscle, which means you get to decide at will whether to keep your eye open or closed.
If all this seems confusing, join the legions of medical students who have to learn about ptosis, lid lag, Horners and the like, and whether or not a patient has myesthenia gravis, is about to die from an aortic aneurysm, or is just bored about reading how the eyelids work.