My common law partner and I bought a house together almost two years ago. Both of our names are on the mortgage. How do I go about taking my name off as we broke up?

The mortgage company is not going to allow you to just take your name off the mortgage to the home. It's not that simple. Your exbf needs to get his own mortgage. In other words a brand new mortgage with just his name on it. If he does not chose to do this, you are going to remain on that mortgage until it is paid off.

I agree, the bank want to protect their investment and removing a name is not in their interest. The same lender may be interested in giving a new loan but the ex will need to qualify on his own assets

Yes, you are going to "sell" your interest of the house to him, this can be for 1 dollar but he will have to get his own mortgage in just his name to pay off the joint one