First, it is indeed really doubtful that Burgess could imprison Death of Endless. As @Valorum pointed out, she is indeed the most powerful of the Siblings, and the only reason that Dream was trapped was because he was weakened. But let's assume for a moment that indeed Death became imprisoned:
Arguably, Endless are not the only force in their realms: when Dream was imprisoned, most people didn't stop dreaming (Constantine for example has endless nightmares), but the dreaming process became a bit chaotic - in extreme cases some couldn't wake up at all (the "sleeping sickness") or were in the constant near-dream, sleepwalking state. One of the Endless - Destruction - at some stage decided to quit and no longer takes care of his realm and doesn't attend to his duties, yet a force of destruction is still present in the world - just he is no longer managing it.
So what would happen if Death stopped taking care of dying? Well, most likely nothing pleasant, as death would become more unpredictable: some people (animals, plants, germs) would continue to live long after they shouldn't, which on the one hand would give them a chance to recover, on another it could turn their lives into a nightmare. Probably the opposing would also be true, making some people die for no apparent reason. And since Death is also a guardian of the Sunless Lands, you could expect a wave of the dead to come back to the world of the living.
Something like this actually happened, when Lex Luthor merged has merged with the Zone Child, becoming nearly omnipotent - he has stopped death as a force. The Death of Endless was... quite happy actually, enjoying a bit of a break.