So I was wondering whether it is still true even when the system is dissipative like a damped harmonic oscillator?

It is true if the dissipative system is Hamiltonian, i.e. if the dissipative behaviour can be described by time-dependent Hamiltonian. For example, one oscillator connected to million oscillators can be described by the Hamiltonian $$ H = \frac{p^2}{2m} + \frac{1}{2}kx^2 - xF(t), $$ where $F(t)$ is the force acting on the oscilator due to other oscillators. Appropriate function $F(t)$ will make the system behave in a dissipative way, but since the sysdtem is described by Hamiltonian, the Liouville theorem is valid.

So I was wondering whether it is still true even when the system is dissipative like a damped harmonic oscillator?

It is true if the dissipative system is Hamiltonian, i.e. if the dissipative behaviour can be described by time-dependent Hamiltonian. For example, one oscillator connected to million oscillators can be described by the Hamiltonian $$ H = \frac{p^2}{2m} + \frac{1}{2}kx^2 - xF(t), $$ where $F(t)$ is the force acting on the oscilator due to other oscillators. Appropriate function $F(t)$ will make the system behave in a dissipative way, but since the system is described by Hamiltonian, the Liouville theorem is valid.

So I was wondering whether it is still true even when the system is dissipative like a damped harmonic oscillator?

It is true if the dissipative system is Hamiltonian, i.e. if the dissipation can be described by time-dependent term in the Hamiltonian. For example, one oscillator connected to million oscillators. However, damped oscillator defined by

$$ m\ddot x = - kx -bv $$

is not Hamiltonian system and the Liouville theorem is not valid for it.

So I was wondering whether it is still true even when the system is dissipative like a damped harmonic oscillator?

It is true if the dissipative system is Hamiltonian, i.e. if the dissipative behaviour can be described by time-dependent Hamiltonian. For example, one oscillator connected to million oscillators can be described by the Hamiltonian $$ H = \frac{p^2}{2m} + \frac{1}{2}kx^2 - xF(t), $$ where $F(t)$ is the force acting on the oscilator due to other oscillators. Appropriate function $F(t)$ will make the system behave in a dissipative way, but since the sysdtem is described by Hamiltonian, the Liouville theorem is valid.