I am sorry to hear your father is not responding to all your encouragement and know how very difficult it is to be on the caregivers side of the fence. My husband responded in a similar manner following treatment, it was like he crawled in a hole and pulled the hole in after himself. First I think your father is not that far out of treatment, and the other problem I found was that patient and family are not mentally prepared for the the slow recovery following treatment. The treatment causes such nasty side effects, and all you can think of is how much better it will be when all this is finished, and then when this doesn't happen as quickly as we thought it would it is quite difficult to handle. Also think how your father coped with stress before he was ill? For my husband it was to avoid it, walk away from it. When the cancer happened, he could not walk away from it and I, nor anyone else could take it away for him. He just could not emotionally handle it. I do think you should get some psyciatric support for him if he will consent to it, and he would likely be put on anti-depressants. Just be aware that it takes up to 4 weeks for the medication to reach its full effect and he may need to try different medications before the most effective one is found.
Is there a support group or could his treatment team have an oral cancer survivor visit him so he can see there is light at the end of the tunnel. Wishing you all strength and courage to face each day.