For Bosons obeying Bose-Einstein Statistics, Why is energy taken to be zero at 0K, while for Fermions the energy is not zero at 0K.
Bosons all collapse to the ground state, since there is no restriction on the number of particles that can occupy a given state. You can assign $0$ the ground state energy, or any other number really since you can't really measure its energy, but only differences between energy levels. But it's customary to choose $0$ (the only time you need to get the ground state energy "right" is when you do GR, since gravity sees everything).
Fermions cannot occupy the same state. There is at most one fermion per state (where by "state" I mean something labelled by all possible quantum numbers, i.e. completely specified). They will build up from the bottom state. So each of them will have a different energy. Since it's statistical physics, you take the average of all the energies and you find that such average is $\neq 0$, even though one particle will indeed have $0$ energy (the one in the ground state) and others will have a larger energy (the ones in higher states).