[Off Oil]

I am guessing it only affects the efficiency of the car, if that. However, it is causing my car not to be able to remote start and that can be problematic sometimes in my climate. So, I should assume they're going to charge me an astronomical price for diagnosing it and then again for something out of warranty aren't they?

If the shutter is stuck closed, it's restricting airflow and therefore it shouldn't be ignored. The only way to know the cost is to actually get a diagnosis and repair quote. Assumptions are a bad thing...