Why is the ionization energy for Hydrogen non-zero?
The ionization energy (IE) of an atom or molecule describes the minimum amount of energy required to remove an electron (to infinity) from the atom or molecule in the gaseous state.


With ionization energy, an electron is not "kicked out" by other electrons, but rather it is "the energy required for the electron to 'climb out' and leave the atom." 

Since the electron "is drawn inwards by positive electrostatic potential," it would make sense to infer that the more "tightly drawn" the electron is to the nucleus, the more energy it would require to "climb out."

More of this can be found on the wikipedia page.