I live with my common law partner but I’m not on the lease…if something happens to him do I have to move out?

Yes.  Even though you may be common law spouses under Ontario Family Law, you do not have equal rights in the home or apartment you live in together.   If your partner is the only person whose name is on the lease, then he is the only one who is considered to be the tenant, in law.  This means he alone has an enforceable obligation to pay rent to the landlord, and in return is the only person with the legal right to live there.