How do you deal with hallucinations?

You mention he has Alzheimer's. Is it truly Alzheimer's or some other dementia? Some present with hallucinations more than others. A proper diagnosis is important.
You MUST take the keys away from him. Yes it is difficult loosing independence but he should not be driving. Use any excuse you can think of. "The doctor said with this medication you can not drive". is a great one and I used that with my Husband. The car is in the shop. Disable the car so he can not start it.
With the dementia he should not be left alone.
Is he by chance a Veteran? If so you might be able to get some help from the VA and depending on where and when he served it might be a little help or a LOT.

Your boyfriend is experiencing delusions, not hallucinations. Someone who is delusional believes something that is not true. He believes you want him out of the house, although you assure him you don't and that you love him. Trying to prove your innocence, however, won't work, his mind is made up. Some delusions are very benign and at times it can be helpful to join their delusion. In your case, it is not! Harmful delusions, such as his accusations, are difficult to deal with. Continue to express your love for him and that you need him. Bring his behavior to his doctors attention.

Some states only require a notice to the DMV that a person's driving is dangerous. This could be a written letter to the dept citing that your LO has dementia. In other states a notice from a physician may be required for the dept to take any action. The dept may then ask the person to come in for a driving assessment. Your goal here is to place the blame for removing his driving privileges on someone else... the doctor or the BMV, not you.

There are medications available to help people who experience hallucinations. Is your boyfriend on any of them? If not, you may want him to ask his Dr. about them.
Your profile says that you work with your Area Agency on Aging, along with an Alzheimer's group. I would think you would have access to all kinds of information that would be helpful in your situation.
And if your boyfriend has Alzheimer's/dementia, he really shouldn't be driving. I know that you're trying to keep the peace, but what if he were to kill someone because he made a mistake while driving? You would feel much worse, than having him mad at you for a little while. You may need to talk to his Dr.(through the patient portal if necessary) about him not driving. The Dr. can notify the DMV, and then you can blame the fact that he can't drive on his Dr.