Connecting an Arduino Nano, Mega and LCD via I2C works but fails when shutting down Mega
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How to setup an I2C bus which stays valid if a slave is powered down or fails?

Basically I2C uses pull-ups to have all lines HIGH until any device bring it LOW. Arduino uses INPUT_PULLUP / OUTPUT+LOW states for this. But if you switch Arduino off (turn off Vcc), internal diodes cut the power on pins to Vcc (which is zero) +diode voltage (which is small), effectively to logical LOW. And I2C see that as the device is activelly sending LOW all the time - so communication is blocked. 

If you just add pullup resistors, you either have them high enought so it not prevent this state or you have them low enought to draw the lines HIGH even when there is LOW on device, which prevents other devices to draw them LOW for communication. You need "cut off" such incactive lines - either physically disconnect (pull wires from connector - such as if you just get the Mega out of socket, or use some relays powered from Mega Vcc, which disconnet them when the power goes out) or electrically disconnect - using some circuits designed for it. There are some already sold or you can create your own to the same efect (if you are skilled in this area - otherwise it is much easier and faster to pay for some already made solution)