One kinda hilarious answer involve quantum mechanics and being able to observe effects "too soon". I mean, I'm not an expert on quantum mechanics, but from what I understand, it's observation that collapses the waveform down. If you did the double-slit experiment, but were able to "see" where the photon would hit before it actually reached the double slit apparatus, I'm not sure the photon would be able to interfere with itself like usual. I have a feeling that a 'precog' looking at the double-slit experiment would never see a diffraction pattern, because the "observation" would be occurring before the photon actually got a chance to interfere with itself.
If that's the case? Then the solution is hilariously simple: just have a whole bunch of slits/dots/whatever in a sensitive box that breaks when you open it. Shine coherent light in one end, and look at the pattern that comes out. If no precogs were around, everyone would just see a mishmash of interference patterns. If a precog was around, it'd instantly change to be dashes/dots/whatever.
Edit: Delving a bit more into the quantum side of things. Basically, everything is a probability wave. How the double-slit experiment works is a photon is beamed at two slits, which it will pass through and arrive at a sensor that measures where the light hit. Intuitively, you'd expect that the photon would have to go through one of the slits and you'd see a 'hit' along two vertical lines. Repeat it over and over, and you'll get a rather weird result: an interference pattern. Basically, the photon is simultaneously going through both slits, and interfering as a wave - against itself!
Now, stage 2 of weirdness. Let's say you rig up something that can detect which slit the photon travels through. You then repeat the experiment. Now, though, the interference pattern disappears. That's because your information 'collapses' that wave down. You've 'erased' the possibility that it went through one of the slits, and because of that, the wave can no longer go through both slits and interfere with itself.
Stage 3 of weirdness. You use the same rig as Stage 2 - and measure which slit the photon travels through. But... now you delete this information before the photon can reach the detector (this is called the Quantum Eraser Experiment.) Now the interference pattern reappears! Why? Because while you measured which slit it went through, you deleted that information - and in effect, restored the possibility that it could have gone through either slit.
So, now, the big question: what's the effect of seeing where the photon lands before it goes through the slit or actually lands? And... ultimately, it's a time travel paradox.
If you observe the photon landing in a way that indicates it went through a specific slit, then you observe something that collapses the waveform down in a way that forces it to go through the specific slit, which in turn prevents an interference pattern from emerging.
If you observe the photon landing in a way that indicates an interference pattern, then you observe something that doesn't collapse the waveform down in a way that forces a specific slit, which in turn allows the photon to travel through both slits, which allows for the interference pattern.
Now, if this was a science site? I'd say, "There's no way to empirically test this; there's no way to answer this." But we're in a worldbuilding site. So you can resolve a time travel paradox however you want. You can say, "So what if they see an interference pattern a bit earlier than the photon actually hits the screen? The photon, being in an interference pattern, doesn't convey information about the waveform that would actually reduce it down." And you can say, "The precog's going to see the photon's final state on the wall based on which slit it went through - because their observation itself is what forces the photon to not interfere with itself."
Anyway, in my opinion, the thing that makes this answer so cool is... ultimately, it's taking something so ridiculously unintuitive and complicated and mindbending... and the resulting device is something a child could manufacture. (Well, not the 'can't open/scan' part.)