Does a LiPo balancer board need switching for external power supply
Base on comercial website comercial website it give this data


Module will discharge  cell which exceed 4.2V
Module will cut-off charging when any cell exceed 4.25V
So for your question


Does this mean if all batteries are full it just starts &quot;burning&quot; power through the resistors for all cells?

The answer is no because it cut-off charging current.

are there reasons why I would need to switch off the power to the module when the batteries are full?

No, the swith already implement in BMS module.

Also how would I typically notice, that the batteries are full?

Of cause, by detect battery voltage. Battery full when it reach 4.2V per cell

I am assuming the 16.8V on the module always supplies the 16.8V independent of the charge of the batteries?

Yes, just like normal battery it supply voltage. If you want to charge its, you need more voltage to force current back in.

Do I need to create a system like in this question

Yes, when you have more than 1 power source for 1 load. 0/16.8V therminal are just like battery lead. You can directly connect when you need to switch power supply.
For second question. Yes diode solution is working but it highly lost and heat up.
Anyway for charging battery you nedd to control charging current. The module not limit charging current. When you put generator directly to battery It's might supply overcurrent and damage battery. If you planned to use buck convertor I strongly recommend to use CC/CV type and set CC to match charging current of battery.