The roof of my mouth has started to hurt when I apply pressure to it. What could cause this soreness?

Palatal Soreness. I would visit with an oral and maxillofacial surgeon and/or ENT, and get thoroughly examined. It could be something relatively benign, or it could be a more serious underlying pathology. Always good to get things checked out sooner than later.
Anything. Without seeing it, one may only guess. There may be infection such as thrush that causes pain. Torus palatinus (bony growth of the palate bone) that is a benign condition usually doesn't cause pain, but may cause discomfort if pushed. If your mouth doens't hurt when you don't push on it, you may as well not push on it and see what happens. It may very well be nothing serious.