We know that the rich man was aware of the poor guy--he even knew his name. I'm pausing here to wonder why I think that ignoring the needs of people whose names we know is different from ignoring those of strangers.
He not only knows the name of Lazarus; he wants to be waited on by him. "I need something. Send Lazarus to help me."
Abraham informs him that the situation is now reversed: the one who had good things now doesn't, and the one who had suffered in his lifetime is now comforted. Moreover, the time to change that is past. The rich man has lost the opportunity to use anything that he once controlled.
The rich man reacts by wanting to ensure that his relatives don't end up the way he has. He begs Abraham, "Send Lazarus to warn my brothers so they won't have to end up the way I have." Abraham reminds the rich man that those brothers have already received sufficient warning because they have access to the Bible. Abraham says that he doesn't think the Bible works for everyone, but that a visitation from someone who has died would.
I would be willing to interpret this remark as a reference to the resurrected Christ, but Abraham's next remark limits that willingness. He says, "If they don't believe the Old Testament, they won't believe the New one either."
Believing in Christ does not mean that we are to cut up and throw away the front part of our Bibles. And if we were to do that, we would miss a whole lot about the necessity of helping the poor. Just saying.