Many A/V receivers have a mode that allows either the last chosen HDMI input, or a particular programmed HDMI input, to just pass through the receiver when the receiver is turned off.
This may not be available in a mid-range unit, but that's the feature to look for.
A good answer is easier with a clear question giving the make and model of everything. "The biggest problem in communication is the illusion that it has taken place." -- G. “Bernie” Shaw