What should I do to unused input parts/features (analog and digital input pins, and power pins) of a video scaler chip?
Regarding the power pins, you almost certainly need to connect them the same way that you would if you were actively using those interfaces on the IC.  In devices that have multiple sets of power pins that are called out as being specifically for different parts of the device, they are quite often still connected to other pins on the device internally--they are called out as being for a specific part because they are most closely connected to that part of the die and will thus supply the bulk of the current to that area.

The exception to this is when different parts of the IC are designed to run at different voltages at the same time, or when the IC supports certain power supplies being shut down while other parts of the device are still active.  The former is common in fast mixed-signal devices (like Ethernet switches/PHYs) and high performance processors or FPGAs.  In these cases, any power pins that are always expected to be at the same voltage may still be connected internally.

In your case, if the DP_VDD (for example) pins are internally connected to the VGA_VDD pins, then leaving the DP_VDD pins unconnected means that the DP interface will still be active and will be drawing current through the VGA_VDD pins.  In the best case, the DisplayPort electronics will be inactive and won't draw enough current to matter.  In the worse case, the DisplayPort electronics will still be active and can draw enough current through the other pins to disrupt operation in other parts of the IC.  In extreme cases, the DisplayPort electronics will draw enough current that you could wind up exceeding the absolute maximum input current per pin on the VGA_VDD pins, or enough current flows across the die that the IC is damaged internally.

Devices that do have internally separate power networks and are designed to operate with one or more of those networks unpowered have to take special precautions wherever signals cross from one power domain to another to avoid problems, so even if the DP_VDD pins on your device are not internally connected to the VGA_VDD pins, this is no guarantee that the device will function properly with the DP_VDD pins unpowered.

So as a general rule, unless the datasheet explicitly states that you can leave those power pins unconnected in your application, you should power them as you would if you were using that interface.  In any case, generally, you should ALWAYS connect the ground pins.

The signal pins are a little trickier.  If the chip requires pullups/downs to operate properly when the DisplayPort cable is disconnected, then you should probably use the same pullup/down configuration in your application.  If it does not require pullups/downs normally, then you probably don't need them either.

If the IC supports configuration via pin strapping or via a digital interface that allows you to completely shut down the unused interfaces, then it's more likely that you'll be able to leave off pullups/downs that would otherwise be required on those interfaces.

If in doubt, it's always a good idea to place the footprints for pullups/downs so that you can make adjustments in testing.  In production, you can then simply fit or not fit whatever parts you find you need or don't need.  Empty PCB footprints are essentially free.