Some viruses (such as bird flu and west nile) can cause tremors and neurological effects like this without any other signs of illness. These types of viruses attack the brain stem, if caught in time with antibiotics, it can recover without any neurological damage. If this is the problem, It may be late considering the severity of the tremors, but it's still possible it can recover, there may be the chance it will always have "the shakes" if there is already unrepairable damage. You would need to treat it right away because left alone, the baby's immune system may not be able to keep fighting and it can cause other issues like organ failure that will ultimately kill it.
I think you have a really good chance since he is eating, drinking and pooping good. I don't think it is a toxicity issue. I would start antibiotics right away. Even if it is not a virus, it won't hurt, there is obviously something going on and it may help.