How to make my mom and sister get along?

This is honestly something that will have to be solved between them. I really don't think there is anything you can feasibly do - aside from encouraging them to talk to each other, or talk to someone who can help them work through it (like a counselor) - to fix this situation yourself. I know from experience, sadly, that sometimes people are simply too stubborn and cemented in their ways to change their mind, and although it is unfortunate, they must ultimately make the decision for themselves, based on their own priorities and values, as to whether they will open themselves up to other ideas or not. 

I'm sorry this is happening to your family - I am actually dealing with some similar issues that came up after my sister came out as gay, and I know how hard it can be when there are people that are accepting and some people who have basically disowned my sister over it. It's hard to accept that some things can't be changed when we want them to, especially when we care about everyone involved. I wish you the best of luck and I really hope your mom eventually comes around. 

I don't exactly know.

Your Mother, even though she has her strict beliefs, has to realize that your sister is an adult who is free to make her own decisions and choices. It is sad that she has distanced herself from your sister. Please talk to your Mum and ask her to reconcile because life is too short for unhappiness and regrets.

You could try

famiy

therapy 

Your mother has no control over your sister. you have to talk to your mother about letting go, she doens't have control, will NEVER have control.

SHE is the problem in this conflict

You can't broker peace but you can act out how you want to be. They are adults.

you cant, all you can do is pray they do

I realize you mean well. Frankly I think separating is a good way to live.

Currently I am looking, for a home to live in. I don't want to live with other people, and it doesn't matter if they are related to me, or not.

If your sister wants to live with her boyfriend, and your mom wants to cut her off then let her. I assume you want to see both of them happy right, so if separating makes them happy then just be happy, for both of them, and get on with your own life.

You can't.  Only your sister and your mother can choose to put aside their differences and play nice.  A couple of thoughts:

1. There is nothing Biblical about cutting "sinful" people out of your life.  We are called to love everyone -- especially the sinner.  While I doubt your mother would see it this way, you might point out that Jesus spent most of his time with people who were sinning in some way or another.  

2.  Your sister made a clear choice knowing that her choice would break her mother's heart.  She doesn't HAVE TO live with her boyfriend. Your sister could not live with her boyfriend.She's chosen to live with him and reject her childhood/parents' religious beliefs.  That's her choice, but she (and you) shouldn't be surprised by the end result.  

3.  You may have to choose sides.  If you stay silent and stick with your mother, you will not have much of a relationship with your sister.  If you speak your own mind and decide to celebrate the holidays with your sister -- and generally actively engage in a relationship with her, your mother will have to make her own decision about how to reach.  

You are never going to change your mom's viewpoint on this. My dad was like this and he never changed. See your sister but do not bring up your sister to your mom and if she brings her up, say mom I don't want to hear about it. Gotta go. It is between them to work it out and I would stay completely out of it.