MCP73831 Li ion battery charger problem with load sharing circuit
I am also using this load sharing circuit and it works for me. You say that the diode is heating while the battery is charged. This means you have 5V at the input and the Mosfet is closed. To help you we need some more information


voltage drop across Diode
voltage drop across the transistor


If you have 5V at the input you should have a voltage drop about 0.25V across the Diode and therefore about 4.7V at the load.
If your Lipo is not fully charged, lets say 3.5V and you Transistor is not blocking, your Output will be pulled down to 3.5V. This means you will have a Voltage drop of 5V-3.5V across the diode. That might be the reason why your diode is heating. Please check that