"On paper" typically means writing something down physically, using pen and paper.
So, depending on what you want to do, the correct phrase will be:
* "Write in the paper" if you're talking about contributing to a newspaper or magazine.
* "Write on paper" if you're talking about physically writing something down.
For example:
* "I'm going to write in the paper about the new park opening."
* "I'll write on paper my grocery list so I don't forget anything."