Ultrasonic cutting with a disc cutter is an advanced machining method for the high-quality processing of Nomex honeycomb core. The machining quality is influenced by ultrasonic cutting parameters, as well as tool orientations, which are determined by the multi-axis machining requirements and the angle control of the cutting system. However, in existing research, the effect of the disc cutter orientation on the machining quality has not been studied in depth, and practical guidance for the use of disc cutters is lacking. In this work, the inclined ultrasonic cutting process with a disc cutter was analyzed, and cutting experiments with different inclination angles were conducted. The theoretical residual height models of the honeycomb core, as a result of the lead and tilt angles, were established and verified with the results obtained by a linear laser displacement sensor. Research shows that the residual height of the honeycomb core, as a result of the tilt angle, is much larger than that as a result of the lead angle. Furthermore, the tearing of the cell wall on the machined surface was observed, and the effects of the ultrasonic vibration, lead angle, and tilt angle on the tear rate and tear length of the cell wall were studied. Experimental results revealed that ultrasonic vibration can effectively decrease the tearing of the cell wall and improve the machining quality. Changes in the tilt angle have less effect than changes in the lead angle on the tearing of the cell wall. The determination of inclination angles should consider the actual processing requirements for the residual height and the machining quality of the cell wall. This study investigates the influence of the inclination angles of a disc cutter on the machining quality of Nomex honeycomb core in ultrasonic cutting and provides guidelines for machining.