After the resurrection of the dead, will our bodies stay alive forever, or will the soul later depart from the body?
The ultimate reward will be for souls and bodies together, at the resurrection of the dead.
There is actually some debate on this issue in classical Jewish writings. Maimonides posits that after the resurrection of the dead, there will come a period when the physical body will return to dust, and the soul will be elevated to a very high spiritual state that cannot be contained in a physical body. This is also implied in the words of our sages, "In the World to Come, there will be no eating or drinking" - we will no longer have physical bodies, and thus will not need physical sustenance.
However, Maimonides' view is a minority opinion. Nachmanides and the great Kabbalists disagree with his view (and in fact, there is some support for the opposing view in Maimonides' own writings.) They believe that the ultimate reward will be for souls and bodies together, at the resurrection of the dead. The body will be elevated to its peak of refinement and completion, and will be a fitting vessel to contain the highest spiritual lights.
Chassidic teachings follow the view of Nachmanides. They explain in the World to Come, the physical body will be so refined that it will no longer need physical nourishment, and will be sustained by spirituality alone.
This view follows the general thrust of Chassidic teachings, which state that our purpose in this world is not to separate ourselves from physicality, but rather to live within the world and elevate it to holiness. This refinement will reach its pinnacle with the ultimate redemption, when the body will in fact reach a higher spiritual level than the soul. Rather than the soul vivifying the body, the body will actually vivify the soul!