Pure conjecture, if you are on an ecclesiastical visa (does such a thing exist?), they could potentially inform the authorities that you are no longer there with their blessing. You might need to leave and come back to get a tourist visa. Depends on where it is, basically.
Billing and rent is all handled by the church. So, if they ask you to leave, you'd technically be squatting, and subject to eviction.
It's unlikely they'd try to evict immediately (which can take up to a month or two in some states), though, as that would be both hugely embarrassing and somewhat expensive. They'd probably just buy you a plane ticket home, or threaten not to buy you a plane ticket home if you stick around. I'm not familiar with non-US landlord/tenant laws, so no ideas there.
Or if you are on a work visa of some sort because you are losing your job. As for paying rent, I presume one pays for the mission on a monthly basis rather than the whole thing up front.