Yes, you need supports on the static structural analysis when inertia relief is turned ON.
I did my own study. I used three methods of supporting a structure to use with Inertia Relief: Weak Springs, Remote Displacement (Deformable) on the left, or a 3 point kinematic mount. For systems A, B, C I applied equal and opposite forces on the left and right edges for the static equilibrium cases. For systems D, E, F, I only applied a force to the left edge, which requires there to be a ridiculously high acceleration to balance that force.
Looking at the above three results, Weak springs is slightly worse than the other two, but less than 0.5% difference from the ideal, which I would call an insignificant difference. In the results below, there is only a force on the left edge. Compared with the results above where the forces were in static equilibrium, there is a huge difference in stress when the forces are not in static equilibrium because now there is a huge acceleration field applied to the solution.
Looking at the above three results, there continues to be no difference in stress as the supports move around for the same unbalanced force on the left edge only, except Weak Springs are slightly off.
So the lesson above is that the applied forces must be correctly applied to get correct stress in an Inertia Relief model.
What happens if you turn off Inertia Relief? When the applied forces are not in static equilibrium, you get a very different stress result. The reason is there is no acceleration applied to drive the reaction forces to zero.
However, when the reaction forces are already zero, then there is no difference in the result.
The lesson here is that when the applied forces are in static equilibrium, you get the same answer whether Inertia Relief is On or Off.