Laura - you don't say what sort of a door frame it is. If it is a painted wooden frame the fix should be quite straight forward.
You will need some sand paper, fine surface filler, and paint to match the original.
Use a sanding block (a piece of wood which you can wrap the paper around to give you a flat sanding surface) and rub back the area chewed by your dog until it is fairly smooth, but most importantly level with or below the wood around it.
Apply the fine surface filler to the teeth mark and chewed portions of the frame so that it fills the marks and is level with or just above the surrounding wood. Allow it to fully harden and dry. When it is dry rub it back with the sanding block until it is level and smooth. Dust it off and the repaint it following the guidelines for the paint you use.
If it is not painted wood, or the damage is greater than teeth marks and slight splintering there are still things you can do but the options are endless depending on the surface etc.