Usually, if the camera supports live view normally, leaving the mirror up in that mode is not likely to cause unusual damage by itself. In general, holding the mirror up is less mechanical wear than repeatedly actuating the mirror and shutter tens of thousands of times.

That said, it can depend on the DSLR design. Some cameras use springs in the mirror mechanism, and long-term tension could contribute to fatigue over time. Others use motor/servo-driven mechanisms and are less affected by that specific issue. So there is no single answer for every DSLR.

The more practical risk is heat. Long live-view/video operation can make a DSLR overheat and shut down; that protection is meant to avoid damage. For a months-long unattended setup, thermal stability is likely a bigger concern than the mirror being up.

If you were considering shooting stills instead, repeated shutter/mirror cycles would create much more wear and could run into shutter-life limits quickly. For this kind of project, continuous live view/HDMI capture is generally the lower-wear approach, provided the camera can manage the heat.