My question is very simple. What components should be removed from the following circuit diagram for this circuit to work only on DC voltage? I'm sure that the bridge rectifier should be removed further than that I do not know.
You can remove everything connected to pin 16, VCC, which is where your DC power will go.
Although, depending on your DC source, you may need a filter capacitor. So, leaving C12 would be a good bet.
If you want to keep the LED power indicator, do so, but you may need to adjust the resistor (R5) value depending on your DC voltage.
From the datasheet it looks your DC supply voltage should be between 3 and 15 V.
And, of course, you might want to keep the power switch.
Basically, placing your DC supply at pin 1 of the switch, would probably work fine (after verifying that the LED resistor is suitable).
Why remove any components? It'll work on DC just fine without any component removals and you won't need to worry about someone putting DC onto the circuit with the wrong polarity because the bridge will protect against reverse polarity.