Would it be possible to fold a paper airplane to an exacting enough degree that a low powered wind tunnel would keep it aloft and relatively static in position?
Not indefinitely. Paper airplanes have no thrust, so they’ll eventually end up at rest relative to the air around them. At that point, you’re going to end up blown up against the back end of the wind tunnel. For a short time? Well, that depends on how short you want.
Well yes I guess I didn’t specify - I mean the imperfect nature of hand folded paper would take you beyond engineering fault tolerance. However, I mean say could it keep aloft longer than a standard thrown by hand airplane?
But it IS a standard thrown by hand airplane, that you’re simply putting in a wind tunnel.
You could angle the wind tunnel slightly downhill.
Stability is a concern. I’d expect that without special measures, the airplane would still drift into one wall or another. But aerodynamics is weird, and strange effects can contribute to stability. It might be possible to design things so that it tends to drift back to the center.
Another problem is the fan. Unless you’re using a stepper motor with digital feedback, the motor speed will fluctuate a few percent. To make this deal work, you’d have to carefully engineer every aspect of the total system.
You could angle the wind tunnel slightly downhill.
Or you could point it straight up and use atumblewing glider.
This. This is analogous to slope-soaring in a glider: the prevailing wind blows uphill over a mountain, and although the glider is descending relative to that air mass, the air mass is moving upward faster than (or at a rate equal to) the sink rate of the aircraft. Under the right circumstances, the aircraft could actually have zero groundspeed and zero vertical speed.
I saw a video of something similar once, but on a much smaller scale. A man was walking forward with a clipboard in his hands tilted 45 degrees down at the front; this produced a breeze up the length of the clipboard. Flying in this breeze was a tiny featherweight glider made of thin balsa sticks and super-lightweight membrane material. By pivoting the clipboard left/right and adjusting his walking speed, he was able to keep the glider centered a few inches above the clipboard.
Do this: On your plane, take off the wings, and the tail, and the nose…just make it into a sphere. Now tilt your downward-sloping wind tunnel even further, and further, until it’s pointing almost straight up. And remove the sides from your wind tunnel. Now you’ve got a ball hovering over a fan like you used to see in appliance stores.
Those things are stable because the pressure difference at the edges of the wind column push the ball back towards the center. If you could shape the wings so that it would turn towards the faster air, you might be able to get a paper plane to hover in a tilted wind column. I think you would have more luck with a paper helicopter.
What if the wind tunnel is on a treadmill?
Better yet, how about placing the treadmill in the wind tunnel and flying the paper airplane over it? Surely it would then fly forever. 
Here you go:
To paraphrase the famous physics joke: “Yes, I found the solution, but it only works for spherical paper planes in a vertical wind tunnel”.
If you allow a string, then basically you’re just looking at a kite. In which case, the answer is yes.