If too little body fat leads to organ failure, then why do people say to avoid things like oil in our diet?

Moderation is key: Some fat in your body is very important as it stores energy you need and protects your internal organs such as the kidneys etc. avoiding oils in your diet is not normal. You need them for certain amino acids which make up the building blocks for your proteins. What you need to do is to minimize saturated fats in your diet.

Separate issues: Excess body fat is MUCH more common than too little. One has to be on an extreme diet (or anorexic) or have malabsorption to get dangerously low body fat. Avoiding oils does not reduce body fat much; eating carbs is the main cause of fat. The reason some say oils should be avoided has to do with the fact that many oils are pro-inflammatory & cause oxidative stress- see http://tinyurl.com/metmg3w