My mother is paranoid and will not discuss her finances or the future. What should I do?

Unfortunately, this is common. You may want to enlist a good friend of your mother's or tell her she should see a financial advisor at her bank or elsewhere. Taking the family dynamics out of it often helps. It won't look so much like you are trying to take over.
Good luck. This can get really sticky - as you've already found out. Third parties are often helpful.
Carol

She's scared of the future, or lack thereof. Putting her head in the sand won't help though. Maybe you could tell her of your plans for the future when you die. That might put her at ease a little. Discuss it matter-of-factually, then transition into her own state of affairs, and what she wants. You could tell her you want to be stuffed and put in the yard. Doesn't have to be so solemn and serious does it?

I like Naheaton's idea ( and my husband and I did this) of talking to your mother about how YOU are planning for your future-----your will, your living will, your "Do Not Rescussitate" form and so on. I could actually see my mother's eyes open wider when I would bring my husband and me into the "aging adult" picture. I would say to go slowly and carefully----Our elders can be very sensitive to the feeling of being "pushed." They feel pushing when we are just helping....but that's the situation many of us deal with. It is delicate, no denying that. But Heaton's suggestion of being matter-of -fact is perfect----that's just what you need to do. Don't be afraid, and kind of plug your nose and jump into the deep end of the pool, so to speak. It's good to have one other person with you when you are having this talk with your mother. Also helpful to go at this in stages.....First, broach the subject, then drop it and let them think about it for a while, then bring it up again and discuss, then drop it again, and so on. This is a process for older people to adjust to new circumstances in their lives, to face reality, and then to take the needed action for their own benefit. Good luck.