Must rice always be cooked with some oil or lard?
Oil or fat is absolutely not necessary to cook rice.

I suspect you may have been taught the pilaf method where the rice is first sauteed in oil or butter, and then liquid is added and the rice is fully cooked.  The purpose of the pilaf method is to add depth of flavor.  When making pilafs, additional herbs, spices, or aromatics (such as onions) are often included.

If you have been taught to add oil without the sauteeing, then the only purpose of it would be to add flavor.  It won't really help prevent sticking, and isn't needed for the rice to cook properly.

There are many methods to cook rice in addition to the pilaf method, and many to do not require any fat.  Simply boiling or steaming in water is completely traditional in many Asian cuisines.

In some cuisines, the crusty rice that sticks to the bottom of the pot or rice cooker is actually highly prized (see Wikipedia's nurunji article).

You can mitigate the crusting of the rice by making sure to use the lowest flame or heat setting that will keep the rice steaming.  I actually prefer to put the whole pot in a moderate oven, rather than do it on the stove top.  

You can also use the pasta method of simply boiling the rice in water, and then draining it when it is done to your liking, to completely avoid the crust.  This will result in less sticky rice since the surface starches will be left in the boiling water.  I understand this method is popular in some parts of India and Pakistan.