Your question is a bit tricky; it easy to find nations or territories whose oil consumption is minuscule like that of the Turks and Caicos Islands, whose oil consumption is less then 1 bbl/day (the U.S consumption is 20.73 million bbl/day!). But the Turks and Caicos Islands have a population of only 32,000.

A better estimate would be to look at Oil consumption for countries per capita.

In this case you can say that the Democratic Republic of the Congo uses less oil then any other country per capita, which is 0.125 bbl/day per 1000 people.

The Turks and Caicos Islands use 3.679 bbl/day per 1,000 people, and the U.S uses 68.838 bbl/day, per 1,000 people.