Crown races can be installed, and likely removed, without a crown race installation tool. On the steel fork I bought for my commuter bike, I used a brass punch to carefully tap the race down onto the fork until fully seated. This was a new fork so I did not have to remove an old race.
On the carbon fork for my road bike, the crown race was aluminum and had a slit in it. I was able to press the crown race in place by hand. This fork had a carbon steerer which may be why the race was split (headset came with the frame/fork).
If you are getting rid of the old crown race and don't wish to save it, use a brass punch to carefully tap it off the shoulder on the fork. Any damage you cause to the race won't be a big deal though you'll likely not leave a scratch. Carefully install the new one with light taps working your way around the race to make sure it's going on evenly. Check your headset tightness after your first few rides and make sure your tension adjustment is still in spec.