Mirror stays up

Check the batteries

Have you got the camera set for mirror lock up. Check and disable if appropriate. When mirror lock is selected the first press of the shutter button moves the mirror up the second press fires the shutter

The mirror has most likely locked up due to the battery being dead. Replace the battery and the mirror should drop down and the shutter will work again.

A few suggestions: The mirror lock up function has been activated. The sensor cleaning mode has been activated, doing this the camera as soon as you trip the shutter the mirror will come up and all other functions of the camera are inoperable. the mirror will stay in that position until it runs out of battery or you turn it off. The part you said about having to removing the battery doesn't make scents for this function though. Sensor cleaning is in the Custom Functions.
I know you said regardless of lenses and batteries how long do you wait before shutting the camera off and removing the battery? I know that's a question and we aren't suppose to answer a question with a question but the reason I'm asking is that the Canon 40D has a long 30 second shutter speed and a "bulb" setting. It is possible that the camera has been set to "AV" and the shutter is timing out, OR it's been turned to BULB and that mirror will stay there until the camera battery is dead before it drops. Couple things to check out before the next thing I'd think about is the amount of shutter actuation. The Canon 40D has a shutter good for 100,000 clicks/actuation. Sounds like a lot but if you are doing time laps or extended video it's very possible to blow the shutter out of these.

Sounds like a loose screw inside shorting out one of the curciut boards! tccamerarepair.com

do you have exact model - this maybe bulb which allows the shutter to stay open for as long as the shutter release is pressed - or mirror lock setting - for macro work to avoid any camera shake when the mirror moves - by locking the mirror

Try pressing and releasing the shutter button. If that doesn't do anything, then turn the camera off and on and press and release the shutter button. If that still doesn't do anything, then turn the camera off, remove and reinsert the batteries (or a fresh set, if readily available), turn the camera on, and press and release the shutter button.

I ran across a Nikon N80 last week which hadn't been used for many moons. I tried the shutter and the mirror promptly locked up. It took about half a dozen tries, but the mirror eventually shook loose or whatever, and the camera now works perfectly.

If the above procedure doesn't do it, then you may have to take it in to a good camera shop for a CLA (clean-lubricate-adjust), the camera equivalent of an automobile tune-up.

Without power/battery the mirror will stay up except in M90 mode or Bulb mode. Rotate the shutter speed dial to M90 and the mirror should return.

By "stayed up", do you mean that the mirror remained in the up position so that you could not see anything in the viewfinder or do you mean that the shutter actually remained open? I do not remember the exact features on your camera, but in general, this is what I would do.

1) Remove the film from the camera and take a picture. Note if you see anything in the viewfinder. If you see nothing, the mirror is in the up position. Verify this by removing the lens and opening the door on the camera back. Look through the lens hole. If you can see through the rectangular hole at the film plane, the shutter is open. If instead you something is obstructing the hole, the shutter is closed. In any case DO NOT TOUCH THE SHUTTER! The shutter is what is covering the hole.

2) If the mirror is in the up position, check to see if your camera has a feature that allows locking the mirror in the up position. Some cameras have this feature to allow use of lenses with very short focal lengths. If this is the case, simply unlock the mirror.

3) If the shutter is the problem, your shutter speed may be set to "T", which stands for time. In the time exposure mode, you press the shutter release once to open the shutter and a second time to close the shutter. Cameras with a "T" setting also have a "B" setting, which stands for bulb. This is a throwback to the old days when it was common to use air-powered shutter releases rather then cable releases. The bulb was a rubber bulb that you squeezed to force air through a tube and push a pin to activate the shutter. The "B" setting keeps the shutter open as long as you hold the shutter release in, but as soon as you take your finger off the shutter release, the shutter closes. Both of these settings are used to make timed exposures. If you find that the problem was that the shutter was set to "T", set the shutter speed to 1/25th second or so and try again. The following sequence should occur; the mirror will flip up and the viewfinder image will disappear, the shutter will open for the prescribed time and close, the mirror will return to the down position and you will again see through the viewfinder.

If this does not work, you may need to take it in for repair.

I'm looking for the source of this same problem. I was able to get my d70 unstuck by:
1. Set to full manual mode.
2. In the menus set mirror lock up to yes and then press shutter release.
3. Turn camera off and then back on

Hope this helps you!