Will iphones shut down before they overheat and become damaged?

Yes, it will shut off if it gets too hot, and display a message saying that.

It is not a shutdown per se, it is deeper than that. A warning comes up, all activity is turned off and the only thing you can do is cool it off. Will not respond to anything else.

Since all systems are down, it won't contribute to any further thermal overload. But most times the overheating is external: you left it in the dash under the sun, for example. So temperature can continue to rise up to the point of damage, but the device can't do anything else to prevent it.