A rise in infant mortality, due to infectious diseases made more prevalent by a decline in resistance due to malnutrition;
A rise in overall mortality due to alcohol; and
A further drop in birth rates as women decide to stop bringing children into the world just to watch them die.
Dropping oil consumption reduces economic activity (and vice versa, in a feedback loop), and that helps the environment while being hard on people.
The impact of oil shortages per se during the Cuban special period has been overstated. Oil consumption fell by just 20% over three years from 1989 to 1992, but the main social impact was economic, from the loss of trade with the Soviet Union. Wikipedia says this:
http://en.wikipedia.org/wiki/Special_Period
The dissolution of the Soviet Union hit the Cuban economy severely. The country lost approximately 80% of its imports, 80% of its exports and its Gross Domestic Product dropped by 34 percent. Food and medicine imports stopped or severely slowed. The largest immediate impact was the loss of nearly all of the petroleum imports from the USSR; Cuba's oil imports dropped to 10% of pre-1990 amounts. Before this, Cuba had been re-exporting any Soviet petroleum it did not consume to other nations for profit, meaning that petroleum had been Cuba's second largest export product before 1990. Once the restored Russian Federation emerged from the former Soviet Union, its administration immediately made clear that it had no intention of delivering petroleum that had been guaranteed the island by the USSR; this resulted in a decrease in Cuban consumption by 20% of its previous level within two years.
http://www.indexmundi.com/energy.aspx?country=cu&product=oil&graph=consumption
The moral of the story is, don't put all your trade eggs in one basket. From this point of view modern industrial countries could weather oil shocks better than Cuba did. For a while...
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