Orthogonal time frequency space (OTFS) modulation technique is promising for high-mobility applications to achieve reliable communications. However, the capacity of OTFS systems is generally limited by the Nyquist criterion, requiring orthogonal pulses in both time and frequency domains. In this paper, we propose a novel multi-carrier faster-than-Nyquist (MC-FTN) signaling scheme for OTFS systems. By adopting non-orthogonal pulses in both time and frequency domains, our scheme significantly improves the capacity of OTFS systems. Specifically, we firstly develop the signal models for both single-input single-output (SISO) and multiple-input multiple-output (MIMO) OTFS systems. Then, we optimize the delay-Doppler (DD) domain precoding matrix at the transmitter to suppress both the inter-symbol interference (ISI) and inter-carrier interference (ICI) introduced by the MC-FTN signaling. For SISO systems, we develop an eigenvalue decomposition (EVD) precoding scheme with optimal power allocation (PA) for achieving the maximum capacity. For MIMO systems, we develop a successive interference cancellation (SIC)-based precoding scheme via decomposing the capacity maximization problem into multiple sub-capacity maximization problems with largely reduced dimensions of optimization variables. Numerical results demonstrate that our proposed MC-FTN-OTFS signaling scheme achieves significantly higher capacity than traditional Nyquist-criterion-based OTFS systems. Moreover, the SIC-based precoding scheme can effectively reduce the complexity of MIMO capacity maximization, while attaining performance close to the optimal EVD-based precoding scheme.