Safe to power up the digital circuit for troubleshooting IC chips?
As happens most of the time the answer is maybe.

If you power up some ICs but not others then input lines that are supposed to have a defined input may be floating and output lines that go to other parts may be connected to an unpowered IC. When you power the input pins of an IC but not the power pins then the device will pull more current than expected from the inputs and due to the protection diodes on the pins will try to power itself from the input lines. This is something you should avoid if possible.

If these are 3.3V or lower parts then the chances are things won't get damaged however it does mean that things won't be in their normal state so just how valid your troubleshooting is is questionable.

Normally you want to power the whole board up to avoid this sort of situation.