As far as I understand it and as per explained by my eye doctor,
If we separate this variable from the other possible ones that can cause eye strain while staring at a monitor like amount of "blue light" (check out the program f.lux), overall brightness, refresh rate, etc., then-
-the variable that can cause eye strain as you explain, "looking at something too close for to long," is caused by overextending the muscles and/or the way in which they contract (some combination of this most likely) in your eyes that causes you to focus on something up close.
So, by looking off into the distance, you are stretching those muscles (changing the way their contracting kinda) and/or giving them a chance to relax for a bit.
I'm quite sure that you and not your environment are actually the true controllers over how these muscles contract.
Try this experiment: Imagine there are mountains behind your computer screen and try to look through your computer screen in order to see them. It may feel unusual at first, and a bit difficult - however if done right you'll be able to feel it.
Here's another experiment: Find some shadow or corner in your room where absolutely no light shines into. It could be a crevice or whatever - you could even make it artificially. It could even be in the depths between the keys in your keyboard. Just as long as it's pitch black. When you look into something that is pitch black, from a visual perspective, neither you nor your eyes know how far that darkness extends - meaning it's left up to you to determine how you focus or not focus on it. I can look between the keys on my keyboard and see pitch black - I know that only extends a a few millimeters if any in reality...but my sight isn't telling me that. For a sight perspective, those depths might really end at the center of the earth or travel on for millions of miles.
- ie. you can change this focus just like as if you were changing the focus on a camera. It is somewhat of a skill, but not too hard of one to acquire.
Now trying to achieve this effect is easier admittedly when there is true distance to look at or into. If you're trying to fake it, I also wouldn't recommend trying to look through your computer screen as in my example above for relief. That was just an extreme example to prove a point - it's possible, but not ideal (and you are still also left with other variables like refresh rate, brightness, etc.).
So as long as you are able to induce this change in eye muscle contractions by actually looking off into the distance, or artificially induce it by tricking your eyes into thinking they are doing so - it doesn't matter as the end goal was accomplished either way. The only thing that differs is the potential intensity of this relief you'll get when trying to induce artificially in contrast to doing so in reality; the artificially part is obviously dependent upon your skill and belief that you can actually do that though.
For the purposes of relieving eye strain, does the perceived increased
distance of looking at an object in a mirror, serve the same purpose
as looking at the same object from twice the distance?
If you perceive it to be true, then your eyes will too - so it would cause the same change in muscle contraction/stretching/relaxation effect that accompanies changes in focus.
After thinking about your mirror trick, I don't actually think it would work. Your perception of the objects will be just as close as if you were looking right at them.