If humanity were to become extinct, do you think another animal species would evolve to take our place? What lessons do you think they'd learn from our successes and failures?
No. Intelligence is not some sort of evolutionary goal. Evolution encourages creatures to better adapt to their environment, that is all. They will simply adapt to whatever environment we leave behind. Non-human species may develop intelligence independently from us, but it will most likely be exceedingly rare, much like how beetles that can shoot fire out of their abdomen are exceptionally rare. This is also a possible explanation for the discrepancy between the theorized frequency of life and the observed lack of it. Life is abundant and resilient, but intelligent life is an oddity.