This paper considers linear precoding for the constant channel-coefficient $K$-user MIMO Gaussian interference channel (MIMO GIC) where each transmitter- $i$ (Tx- $i$) requires the sending of $d_{i}$ independent complex symbols per channel use that take values from fixed finite constellations with uniform distribution to receiver- $i$ (Rx- $i$) for $i=1,2,\ldots, K$. We define the maximum rate achieved by Tx- $i$ using any linear precoder as the signal-to-noise ratio (SNR) tends to infinity when the interference channel coefficients are zero to be the constellation constrained saturation capacity (CCSC) for Tx- $i$. We derive a high-SNR approximation for the rate achieved by Tx- $i$ when interference is treated as noise and this rate is given by the mutual information between Tx- $i$ and Rx- $i$, denoted as $I[\underline{X}_{i};\underline{Y}_{i}]$. A set of necessary and sufficient conditions on the precoders under which $I[\underline{X}_{i};\underline{Y}_{i}]$ tends to CCSC for Tx- $i$ is derived. Interestingly, the precoders designed for interference alignment (IA) satisfy these necessary and sufficient conditions. Furthermore, we propose gradient-ascent-based algorithms to optimize the sum rate achieved by precoding with finite constellation inputs and treating interference as noise. A simulation study using the proposed algorithms for a three-user MIMO GIC with two antennas at each node with $d_{i}=1$ for all $i$ and with BPSK and QPSK inputs shows more than 0.1-b/s/Hz gain in the ergodic sum rate over that yielded by precoders obtained from some known IA algorithms at moderate SNRs.