How do I replace the hard drive on my MacBook Pro?

The 2016 15" MBP does not have replaceable internal storage. It is soldered to the logic board and cannot be removed.

You might consider installing macOS and your user files on an external SSD and use that to startup and run the computer. Install macOS on an external storage device and use it as a startup disk - Apple Support

You can check this website for additional info and guidance: OWC Install Videos - eshop.macsales.com

That's unfortunate, but logic boards do fail.

And no, without a working logic board you cannot boot the computer from any startup drive.

When you replace the computer, you would be wise to take advantage of the Time Machine backup application. Apple provides it preinstalled on the computer. Enabling it requires only that you provide an external drive of sufficient capacity to do the job. (2x-3x the capacity of the internal drive).

Back up your Mac with Time Machine - Apple Support

Don't lose your files again.

That will depend on the model year of the MBP.

Newer models cannot be upgraded.

Please provide us with that info and someone can assist with more guidance.

Find that info by clicking the Apple menu () at the top-left of your screen and then select About This Mac. (Note- for privacy and security reasons, please never post your serial number.)

If the logic board is toast, then replacing the hard drive would have zero effect.

Are you sure they meant the computer logic board and not something else?

The logic board is the entirety of the computer internally and without it, the computer simply will not work regardless of the hard drive.