Not really. The city certainly cannot tell a Church it will lose its religious designation. What it can do is tell the Church if the use of a building changes it must meet the building, accessibility, life safety codes for that occupancy. The changes required would be more substantial if the building was expected to hold a large number of people.
It seems like overkill but what happens if a fire breaks out and there are not enough exits? You can be sure the city would face scrutiny for not requiring needed upgrades.
What happens if fire breaks out during services?
The building seems adequate to hold a lot of sitting people during a worship service. Until the number of shelterees gets up near the number of members in attendance, fire safety should already be good.
I think the issue comes more with sanitation. I suspect restrooms are inadequate for overnight stays but OK for two hour visits.
Don't get me wrong here -- I am not in favor of churches providing shelter for the homeless unless it is in a separate, well managed building. There are just too many bad things that can happen when a church gathers up a significant number of mentally challenged drug addicts and gives them free run of the place.