If I put some E.coli in a petri dish filled with limited nutrition in room temperature and forget them for months or years. What will happen in the final after they overgrowth and used all nutrition left? Will they die or just stop dividing?
They will die, eventually.
The requirement for energy is a thermodynamic constraint; maintaining cellular structures, repairing molecules that spontaneously damage etc takes some energy, and E. coli is not able to reach a level of quiescence sufficient to survive for years. However some cells might survive longer than expected if they enter low-metabolic activity states; that would not be enough to survive years in a petri dish, though.
Note that other species may be able to survive, through special stress-induced processes such as sporulation or cryptobiosis.