Is it possible to turn an Android tablet on or off over a wired connection such as USB? I'm looking to use an Android tablet as a carputer, yet one problem I have is how to get the tablet to turn on and off when I turn my car on and off. With most carputers people will use special automotive power supplies to solve this problem. 

What I want to know is if there is any way I can simulate pressing the power button, but through one of the wired interfaces such as the USB connection? Or would my only option be to open up whatever tablet use up and solder extension leads to the power button contacts, and then figure out some way to simulate a power button press when my car starts?
The "power" button on Android tablets does not behave like that on a computer.   It is really a control for whether the display is on or off, not the tablet, which is generally always on.   You  may be able to get what you want by setting the "Stay awake while power is connected" setting under developer options, and setting the screen timeout to a short value under display settings; if you connect the charging cable to a power source that goes on and off with the ignition, you should be good to go.

If the standby time of your tablet is not enough to handle your typical time with the ignition off, you may want to use an app like Tasker or Locale to set additional rules, like turning Wifi or cell connection off when the car's power is off.   If there's data you want to sync overnight, you could turn Wifi on (for example) an hour before you leave for work.

You may also want to disable the screen lock to minimize distractions while driving, but keep in mind the implication if you have any personal information on the tablet.