There is an important distinction between Bose-Einstein condensation of atoms and quasiparticles, which I will describe below.
Bose-Einstein condensation of an ideal (atomic) Bose gas
For bosons, chemical potential $\mu$ has an important restriction: $\mu<E_{0} \equiv 0$ where $E_{0}$ is the ground state energy. Otherwise, the occupation number $\mathcal{N}(E)$ will have a negative value, which is not physical. The total particle density $n$ is calculated using the density of states per volume $\mathcal{D}(E)$ and the occupation number $\mathcal{N}(E)$ following the Bose-Einstein distribution:
\begin{align}
n &= \int_{0}^{\infty} \mathcal{D}(E)\mathcal{N}(E)dE\\
&= \frac{1}{V}\frac{1}{e^{-\mu/k_{\textrm{B}}T}-1} + \int_{0}^{\infty} \frac{1}{(2\pi)^{2}}\left(\frac{2m}{\hbar^{2}}\right)^{3/2}\frac{\sqrt{E}}{e^{(E-\mu)/k_{\textrm{B}}T}-1}dE
\end{align}
Suppose we hold $T$ constant and increase the particle density $n$ by adding particles to the system (note that $n$ is an increasing function of $\mu$). For density to increase, we must correspondingly raise the value of $\mu$. In the upper limit $\mu \rightarrow 0$, the particle density becomes
\begin{equation}
n \simeq \underbrace{-\frac{k_{\textrm{B}}T}{\mu V}}_{n_{0}} + \underbrace{\frac{2.612}{\lambda_{\textrm{th}}^{3}}}_{n_{\textrm{th}}}
\end{equation}
where $\lambda_{\textrm{th}}$ is the thermal de Broglie wavelength. The first term (ground-state density $n_{0}$) increases indefinitely as $\mu\rightarrow 0$, while the second term (excited-state density $n_{\textrm{th}}$) approaches the finite limit. Therefore, the ground state can accommodate excess particles that all the excited states cannot take. After this point, any added particles must go into the ground state and form a Bose-Einstein condensate. The chemical potential is then expressed as
\begin{equation}
\mu = -\frac{k_{\textrm{B}}T}{(n-n_{\textrm{th}})V} = -\frac{k_{\textrm{B}}T}{N_{0}}
\end{equation}
where $N_{0}$ is the number of particles in the ground state. Note that the chemical potential is exact zero only in the thermodynamic limit, where the total number of particles $N$ is infinite. In a realistic system, the chemical potential is always non-zero.
Bose-Einstein condensation of photons and phonons
However, there are systems where the chemical potential is intrinsically zero even at high temperatures and low densities. The zero chemical potential $\mu = 0$ means that its conjugate variable, number of particles $N$, is not conserved. In other words, if particle-number-changing interactions are dominant, there is no free energy accompanied by adding or removing a particle. The most common example is photons, where they can be absorbed and emitted by matter; hence, the number of photons can vary without the cost of energy. The number of photons is constantly and automatically being adjusted to the Planck distribution
\begin{equation}
\mathcal{N}(E) = \frac{1}{e^{E/k_{\textrm{B}}T}-1}
\end{equation}
which is the Bose-Einstein distribution with zero chemical potential. Using the density of states $\mathcal{D}(E)$ for massless particles, the photon density becomes
\begin{equation}
n = \int_{0}^{\infty}\mathcal{D}(E)\mathcal{N}(E)dE = \int_{0}^{\infty}\frac{E^{2}}{\pi^{2}\hbar^{3}c^{3}}\frac{1}{e^{E/k_{\textrm{B}}T}-1}dE \simeq 2.404\frac{(k_{\textrm{B}}T)^{3}}{\pi\hbar^{3}c^{3}}
\end{equation}
The same holds for acoustic phonons with linear dispersion, except for the light speed $c$ being replaced by the sound velocity.
Note that this satisfies most of the conditions of Bose-Einstein condensation. The density of states vanishes for the ground state $\mathcal{D}(E\rightarrow 0) = 0$, so the number of particles has a finite limit given by the above integration. Also, the ground-state occupation $\mathcal{N}(E=0)$ is infinite. However, the particle density in the ground state is
\begin{equation}
n_{0} \propto \lim_{E\rightarrow 0}\frac{E^{2}}{e^{E/k_{\textrm{B}}T}-1} = 0
\end{equation}
unlike the previous case for atoms where the ground state density diverges as
\begin{equation}
n_{0} \propto \lim_{E\rightarrow 0}\frac{\sqrt{E}}{e^{E/k_{\textrm{B}}T}-1} = \infty.
\end{equation}
The ground-state density $n_{0}$ for photons is never comparable to the thermal photon density $n_{\textrm{th}}$, even though the ground-state occupancy is infinite. Furthermore, any added (removed) particle will disappear (reappear) so that the free energy is minimized, i.e., the chemical potential is zero. The only way to change the average number of particles is to change temperature. Therefore, Bose-Einstein condensation cannot occur in systems without a particle-number conservation.
Furthermore, as noted by the other answer, the ground state of a photon (or an acoustic phonon at $\Gamma$ point) has zero energy except for vacuum fluctuation—there is no excitation at all. The zero energy $\hbar \omega = 0$ essentially means that there is no photon (lattice vibration), and the number of photons (phonons) in the ground state is zero by definition. This comes from the shape of the dispersion.
But what if you engineer a photonic (acoustic) band gap structure so that there is a finite minimum of the photon (phonon) dispersion? Even in this case, and even if the density of states scales as $\mathcal{D}(E)\propto\sqrt{E}$ as in atomic case, another important point should be addressed: quasiparticle timescales.
Bose-Einstein condensation of quasiparticles
Quasiparticles, in general, will disappear after a long time and the number of quasiparticles is conserved only at a certain timescale. As mentioned above, the particle-number conservation is necessary to have a non-zero chemical potential and Bose-Einstein condensation. Therefore, the comparison between the lifetime and other timescales is important.
Quasiparticle lifetime ($\tau_{\textrm{l}}$) has two components: radiative lifetime ($\tau_{\textrm{r}}$) and non-radiative lifetime ($\tau_{\textrm{nr}}$). The former is related to the interaction with photonic modes, while the latter is related to the interaction with other particles such as phonons or defects.
On the other hand, interaction time ($\tau_{\textrm{int}}$) can be classified in several ways. One way is based on the type of particles that scatter: self-interaction ($\tau_{\mathrm{self}}$) and interactions with bath ($\tau_{\mathrm{bath}}$). Another way is to use the number conservation as a criterion: number-conserving interactions ($\tau_{\mathrm{c}}$) and non-number-conserving interactions ($\tau_{\mathrm{nc}}$).
These timescales are typically inter-related. For example, $\tau_{\mathrm{self}}$ can include both $\tau_{\mathrm{c}}$ (e.g. elastic two-magnon collision) and $\tau_{\mathrm{nc}}$ (e.g. second-harmonic generation of photons). As another example, exciton-phonon interactions can take away the energy of excitons but do not destroy excitons ($\tau_{\mathrm{c}}$), while magnon-phonon interactions can destroy magnons by emitting phonons ($\tau_{\mathrm{nc}}$ and $\tau_{\mathrm{nr}}$). For that reason, it is not trivial to make a single statement for the condition of quasiparticle Bose-Einstein condensation. Nevertheless, the following condition must be satisfied:
\begin{equation}
\tau_{\mathrm{c}} < \tau_{\mathrm{l}} < \tau_{\mathrm{nc}}
\end{equation}
The first inequality is the thermalization condition: quasiparticles need long enough lifetime to thermalize themselves ($\tau_{\mathrm{self}}$) or to thermalize to bath ($\tau_{\mathrm{bath}}$). The second inequality implies that the quasiparticle can be treated as a thermodynamic particle if the non-number-conserving scattering time ($\tau_{\mathrm{nc}}$) is longer than the lifetime.
For example, consider the case where some non-thermal quasiparticles are added to an equilibrium system. The system is perturbed from the original equilibrium Bose-Einstein distribution. After $\tau_{\mathrm{c}}$, quasiparticles will have a new distribution with non-zero chemical potential and well-defined temperature, which can be equal to or different from the bath temperature. It is determined by whether the nature of the number-conserving interactions is dominated by $\tau_{\mathrm{self}}$ or $\tau_{\mathrm{bath}}$. If the quasiparticles decay after this timescale but before $\tau_{\mathrm{nc}}$, then the system can be steadily pumped to replenish quasiparticles—called a driven-dissipative system. However, if $\tau_{\mathrm{nc}}$ is shorter than $\tau_{\mathrm{l}}$, the distribution is adjusted to the equilibrium distribution with zero chemical potential, and the quasiparticle will always, at the end, have zero chemical potential. For photons, $\tau_{\mathrm{nc}}$ is typically much shorter than $\tau_{\mathrm{l}}$ so that the number of photons is quickly adjusted to the Planck distribution. For atomic condensates, three-body collision time $\tau_{\mathrm{nc}}$ sets the upper limit of the particle density.
