We present two-dimensional resistive magnetohydrodynamic simulations of line-tied asymmetric magnetic reconnection in the context of solar flare and coronal mass ejection current sheets. The reconnection process is made asymmetric along the inflow direction by allowing the initial upstream magnetic field strengths and densities to differ, and along the outflow direction by placing the initial perturbation near a conducting wall boundary that represents the photosphere. When the upstream magnetic fields are asymmetric, the post-flare loop structure is distorted into a characteristic skewed candle flame shape. The simulations can thus be used to provide constraints on the reconnection asymmetry in post-flare loops. More hard X-ray emission is expected to occur at the footpoint on the weak magnetic field side because energetic particles are more likely to escape the magnetic mirror there than at the strong magnetic field footpoint. The footpoint on the weak magnetic field side is predicted to move more quickly because of the requirement in two dimensions that equal amounts of flux must be reconnected from each upstream region. The X-line drifts away from the conducting wall in all simulations with asymmetric outflow and into the strong magnetic field region during most of the simulations with asymmetric inflow. There is net plasma flow across the X-line for both the inflow and outflow directions. The reconnection exhaust directed away from the obstructing wall is significantly faster than the exhaust directed towards it. The asymmetric inflow condition allows net vorticity in the rising outflow plasmoid which would appear as rolling motions about the flux rope axis.