As for what could go wrong--data in different projections may not share the same origin; therefore, mathematical calculations between such layers could be permitted without the software first interrogating the data for potential differences in origin, or with a view to making appropriate adjustments if necessary. Without the same origin, alignment issues are guaranteed.
The issue in Arc, I believe, is that not all routines are created equal. Some routines will defensively check for differences in projection, first, and either warn you or try to accommodate such differences if possible. However other routines do not check, do not accommodate, and blindly proceed under the assumption that the analyst already protected the routine against mismatched inputs.
Ultimately, Arc is being courteous by warning you that there might be a problem if you proceed, because at some more abstracted, outer ring of the software, it sees what may be an unintended operation about to occur. ..at the very least, it's seeing a bad practice unfolding, and as such it's nagging you. :)
@0kcats makes another good point with respect to performance sacrifices.
This being said, is it such an obstacle to translate both datasets into the same projection before running an analysis?