It is well-known that irreversible MCMC algorithms converge faster to their stationary distributions than reversible ones. Using the special geometric structure of Lie groups $G$ and dissipation fields compatible with the symplectic structure, we construct an irreversible HMC-like MCMC algorithm on $G$, where we first update the momentum by solving an OU process on the corresponding Lie algebra $\mathfrak{g}$, and then approximate the Hamiltonian system on $G \times \mathfrak{g}$ with a reversible symplectic integrator followed by a Metropolis-Hastings correction step. In particular, when the OU process is simulated over sufficiently long times, we recover HMC as a special case. We illustrate this algorithm numerically using the example $G = SO(3)$.