According to Adam Smith, humans are self-interested individuals. If this is true, why does he argue that people should reinvest their wealth? How does this benefit a self-interested individual? Does morality play a role here? What role does productive and unproductive labor play here precisely?
Economics have evolved considerably since Adam Smith and his ideas have been refined. I can think of two approaches to your question:
Why would a self-interested individual re-invest wealth?
1) Financial perspective
Whenever you invest money instead of spending it now, you get paid interest. If that interest is higher than the satisfaction from spending the money now instead of later, then re-investing is in your self-interest.
2) Game-theoretical and behavioral perspective
A self-interested person wants to maximize his or her payoffs. Payoffs can differ from monetary outcomes. If I re-invest my 1000 dollars into bonds for a year I can get 1050 dollars at the end of the year. But if I give 1000 dollars to charity, I may get a lot more satisfaction from that, provided that I am an altruistic person. So if try to measure my payoff from the charitable donation, it may be worth the equivalent of 2000 dollars to me. It is in my self-interest to donate instead of re-investing then. The same reasoning can be applied to saving and spending. If it gives me pleasure to hoard money then I will choose to re-invest rather than spend. Using this approach you can justify any behavior as self-interested or rational.
You can take financial perspective to explore how productive and unproductive labor relates to your question, but if you take more general game-theoretical perspective, then it will be difficult for you to even define "unproductive labor" as even piling money and setting it on fire may be considered productive in some situation. But I would take the latter. After all we should study economics to better understand the world in all its complexity, not to take the easier path.