First off, just to be clear, the chemical potential being equal to zero is different from not having a chemical potential at all (e.g. a photon gas)?
Now: physically, what does having chemical potential $\mu=0$ mean for a gas?
A Bose-Einstein condensate exists below a critical temperature $T_c$, at which $\mu$ hits $0$. What's the connection between chemical potential and Bose-Einstein condensation?
Best Answer
When the chemical potential is 0 the extra free energy needed to add or remove a particle to the system is 0(i.e $\mu=\frac{dA}{dN}=0$. So particles can leave and enter the system without changing the (free) energy.
In A BEC all particles have condensed to the ground state of the system. Particles entering or leaving the system will be added to the ground state or leave from the ground state. If your particles are non interacting the only contribution to energy is Kinetic which is monotonically increasing from 0 with respect to momentum. This means your ground state will have 0 energy and adding or removing a particle to the ground state adds or subtracts 0 energy so $\mu=0$.