How do I convince my father to go to a neurologist to check out his tremor?

We don't bite. It's not a bad experience to see us for tremor. We will listen and ask a few questions, do an exam which is pretty easy for him to do, and probably look at his med list and order a thyroid blood test.
For Peace of Mind!! Simply tell him this could be just essential tremor, which is very treatable. Think catherine hepburn. On the other hand, this could represent early parkinson's disease, especially if it is on just one side & associated with slowness of movement & muscle stiffness on the same side. Decreased natural arm swing on the same side as the tremor can be a dead give-away. If so, he should pursue rx now!