What happens to the heart when an artery gets smaller?

You did not specify which artery. If it is one supplying the heart, it will affect the heart's efficiency. If it becomes completely clogged, part of the heart muscle will die unless the blockage is relieved.

If it is an artery to a limb or organ, the heart will have to work harder, and the amount of blood it can push through the artery will slowly become lessened. Eventually, there will be problems at the other end, and perhaps with the heart as well.