Question: The HX2LP datasheet states a default state of '1' for the OVR# pins. Do they have an internal pull-up resistor or do we need an external pull-up if the pins are not used?
There are no internal pull-ups on the OVR# pins of HX2LP. They are set to inputs by default (on power-up) and do not need to have an external pull-up if you are not using the respective port at all. The OVR# is an input (active low) used to detect over-current and it should be setup to default to a state of '1' (so that the port is not marked or detected as over current).
If you are not using the port, you do not need to use an external pull-up, as you don't care about the status of this unused port.
If you are using this port or there is a possibility of using this port, you will need an external pull-up to set the state to '1' (no over current detected on the port).
In any case having an external pull-up (not leaving it floating) is always a good (recommended) idea regardless of whether the port is configured to be used or not.