Since kirigami is folded as it is being created, the finished product will be heavily creased. This is OK if you are doing it for fun and will be taping them to windows for holiday decorations. But if you want to frame your work or display it more neatly, it will have to be pressed.
Papercuts are very delicate, so you never want to place an iron directly on the paper. You are more likely to tear it than scorch it. Open your papercut and place it on a piece of heavy white cardstock paper. (Don't use colored paper. You know why.) Press the folds open with your fingers, dabbing instead of smoothing. Then cover your work with another piece of white cardstock.
Now you are ready to use heat to press your work flat. I use a commercial dry mount press, set on medium heat for about five minutes.
You can use a household iron with these caveats:
2. Place the cardstock with your papercut on the wool and press straight down. Do not slide the iron over the paper or use a swirling motion. This will cause the cardstock to shift slightly and you could be pressing wrinkles into your work.
3. Never use steam! Make sure there is no water in your iron and the steamer is turned off. Use a hot setting but make sure it is dry.
When taking your work off the white cardstock, try not to pick it up with your fingers. Instead, let it slide off one piece of paper onto the next surface.
Be brave. Have fun!