If billionaires stopped spending what would the economic effect be?
If one person decides to reduce his lifetime consumption by (say) \$100, then someone else (or many someone else's) is made richer by \$100.  For example, that $100 might be left to a single heir.  Or at the other extreme, the person might decide to burn a \$100 bill in which case the benefits are spread to everyone who holds money.  

When people are richer, they can decide to consume more, or to work less, or some of each.  Because income elasticities of labor supply are empirically quite small, you'd expect most of this to come out in increased consumption.