Push the Technic pin in a beam hole, then insert a rod (antenna or minifig tool handle) into the hole in the pin from the other side. The rod will prevent the "lips" of the pin from closing in, therefore preventing the pin from leaving the beam. Then you can pull on anything attached to the other end of the pin.
This example below shows the removal of a pin from a beam:
I figured out a solution while waiting for answers. The tool I used for it was a universal T20 bit which most people have or can easily get access to. I took some documenting pictures of it which I would like to share here.
EDIT: Use a plastic LEGO rod as zovits and others recommend. If you don't have that, use metal tools as a last resort. And if using a metal tool, use Allen a.k.a. Hex key instead of Torx. as PlasmaHH commented.