Common formulations to consider collision avoidance in trajectory optimization often use either preprocessed environments or only check and penalize collisions at discrete time steps. However, when only checking at discrete states, this requires either large margins that prevent manipulation close to obstacles or dense time discretization increasing the dimensionality of the optimization problem in complex environments. Nonetheless, collisions may still occur in the interpolation/transition between two valid states or in environments with thin obstacles. In this work, we introduce a computationally inexpensive continuous-time collision avoidance term in presence of static and moving obstacles. Our penalty is based on conservative advancement and harmonic potential fields and can be used as either a cost or constraint in off-the-shelf non-linear programming solvers. Due to the use of conservative advancement (collision checks) rather than distance computations, our method outperforms discrete collision avoidance based on signed distance constraints resulting in smooth motions with continuous-time safety while planning in discrete time. We evaluate our proposed continuous collision avoidance on scenarios including manipulation of moving targets, locomanipulation on mobile robots, manipulation trajectories for humanoids, and quadrotor path planning and compare penalty terms based on harmonic potential fields with ones derived from contact normals.