Do we die when we sleep?

In the context of the conventional physical worldview, there is no evidence to indicate that the life of an individual ends during any phase of sleep. The human brain and body are very much alive during all phases of sleep.

I suspect, however, that the questioner may be looking for an answer from the perspective of a non-physical worldview, e.g., one which takes the immediacy of pure subjective awareness as fundamentally real instead of an objective physical world. From that perspective, because the experience of a separate, individual (empirical) self completely vanishes during deep sleep, in this sense one could say that the experience of self 'dies' when going to sleep, and is then 'born' again when dream or waking states arise.

This kind of death and rebirth is most obvious between the transitions between deep sleep (when there are no objects of awareness at all) and states of waking or dreaming, but it is not limited to that transition. In fact, it is happening moment-to-moment during waking experience, as the experience of self is repeatedly reconstructed. If this is recognized with sufficient clarity and depth of insight, it becomes evident that the experiential self is not an experience of some real, enduring entity. In that sense, there is no real 'self' to actually die or be born.