Spending a fortune on getting someone in to do this or risk losing your rental deposit over a job that takes 30mins-2hrs to fix?
1) Remove excess from around hole with scraper/hands
2) Sand hole
3) Place mesh over hole (if larger hole you might want to fill with expanding foam before applying mesh)
4) Fill with plasterboard filler (make with toothpick if the wood grain is obvious)
5) Sand and then paint
6) Further sanding may be needed
7) Painting the whole door may be needed if the colours are off etc.