INA3221 loses i2c communication when power supply is connected

Well actually an I2C bus pins are active LOW. That is they normally sit at a high position and then in operation they are pulled LOW.

The problem is that when they are sitting at the active state, this must be compatible with the maximum voltage of all the other devices on the same bus.

I am afrade that there is still a problem with your schematic. Tom said:-

Now you simply have a box that says controller. I am assuming this is an Arduino but what sort of Arduino?

Words do not cut it in electronics because they are ambiguous. When you say

Exactly what pin are you disconnecting? Is it the VPU or is it the VS pin?

Can you post a link to where you bought it from please.

This guy had little in the way of technical response.
Getting it to work
He was looking to use it to measure high sided loads.

This thread showed how this was fixed
fixing the ina3221 breakout board
It looks like there are different versions of this board, floating about, and if you have not used a reputable suppler then they are unlikely to know what board you have.

But I think you have a different problem.

Wire it up properly, with the VPU connected to the positive of your power supply and measure the voltage on each one of the I2C lines (SCL & SDA). What is the reading you get?
Then disconnect all power, and measure the resistance between each I2C line and the VS line. Remember never try and measure resistance when anything is powered in the system.

The data sheet to the chip is at
Data Sheet
But remember the data sheet is not the circuit you have on that board. It has a "typical" circuit, and it is likely to be this, because most makers of cheap breakout boards just follow the circuit.

However, please remove all motors until you get this going, just use an LED with a suitable series resistor to keep it safe, and a small parallel resistor so that it draws current for you to measure. Motors are difficult because they generate interference that can upset / reset your Arduino and they always need a kick back diode on them.