Well, carbon dating would generally be for things that contain... carbon. Organic substances - things that were once alive and got their carbon from the air in their environment. The radioactive isotope of carbon is produced from the interaction of cosmic rays (rays from the sun) and nitrogen in the atmosphere. C14 atoms end up being produced in a secondary process. IIRC, the reaction leaves the C14 atoms "hot", and they quickly react with oxygen to make carbon monoxide, and then carbon dioxide. Due to variations in the sun's output, the process isn't quite at a constant rate, though close enough. Because the age of certain artifacts are known precisely, the carbon dating scale can be very fine tuned for precision.
Anyway, that's not the only way to make carbon 14. I suppose, if you have a nuclear reactor laying around, or some atomic weapons, you could artificially increase the amount of C14 in an environment, stick sheep into that environment long enough for it to take up a lot more C14 than it would in the natural environment, then harvest the wool and make some clothing.
BUT, the outcome of that would be that in 5000 years, someone would say, "wow, according to carbon dating, this thing is like 2 weeks old. It sure looks older than that." If you carbon dated it now, because of the higher than expected level of C14, the lab results would be something like, "weird, the computer just spit out that this blanket was made from wool harvested 5000 years in the future. That doesn't make sense.
So, very expensive to fake - in that direction. What about the other direction? I'm not sure what you would be able to do to a material already containing C14 that could knock it down to C12, without it being detected that you did it. But, I suppose bombarding it with neutrons, or alpha particles or some other radiation might be able to do something.