2012 egr cooler leaking

Take a look at this thread, https://priuschat.com/index.php?posts/3360511

How many miles on the car?

 

I got kinda lost after that. What's the miles on it?

 

Unplugging the egr valve is highly unlikely to stop white smoke especially if was antifreeze (eg coolant).

Unplugging the egr when the egr valve is closed (normally closed at low rpm and shutdown) will create a check engine light but not much else you will notice. I might do it except the check engine light on all the time will mask other problems. Other hybrid issues have their own warning lights. The overheating warning light is separate but often just flashes once or twice when it's all over except the crying.

Yes there is coolant in the egr cooler but it is a stainless steel heat exchanger with the coolant separated from the exhaust gas flow. Even if your egr cooler was leaking internally (something that is almost unheard of), closing the egr valve would simply stop coolant flow to the intake manifold. A leaking egr cooler would continue to leak coolant directly into the exhaust system. This would quickly empty the engine coolant reservoir (passenger side) and soon cause engine overheating.

The frequent cause of coolant burning is coolant entering a cylinder through a head gasket leak. These leaks can be infrequent at first but are almost always accompanied by a severe startup rattle when they happen.