This study numerically investigated the melting performance of a paraffin/copper foam composite phase change material (PCM) heat storage unit with a rectangular encapsulation under the effect of different inclination angles. Taking the coupled heat conduction and natural convection into consideration, the transient mathematical models were solved numerically, and the results were validated using experimental data from the literature. The results show that although the ligaments of the metal foams had a strong suppressive influence on natural convection, the composite PCM systems with different tilt angles exhibited different thermal behaviours and melting performances. Natural convection was completely suppressed in the horizontal cases (θ = 0° and 180°) and tended to be stronger for tilted container systems, especially for the vertical case. Due to the natural convection, a heat transfer dead zone appeared at the bottom of the tilted container. The boundary heat fluxes of horizontal cases were highest in the later stage of melting and the total melting time was shorter than that of the other tilted cases. Furthermore, the results indicated that the melting rate of the composite PCM system could be further increased by adopting a larger Rayleigh number and metal foam with smaller pore density.