How can I make an acidic pen to burn paper on writing on it?
You can pour anything into your pen. Is it a good idea? Most likely not. Is it dangerous? You bet! Is it a good idea to play with? NO! Whether it is legal or not to create such an object, you'll have to check with legislation in your country.

If you want paper to ignite on touch with this pen, you would need some truly strong acid (and probably undiluted). I'm certain that if this substance has the capability to burn paper, it would also corrode away the feed and nib of the pen. In the worst case, it will even eat through the ink container and leaking everywhere. In any way, this is no viable option to cut paper in a controlled fashion.

Is there a safer method? Well, there would be laser-assisted cutting of paper, but the good ol' knife or saw would still be cheaper.