Why can't people have "stirrings" in Jonas's society?

There is no explicit reason given; however, we can deduce the answer to this based on other things we know about the society.

We know that the stirrings are closely regulated by the community. In Ch. 5 Jonas gets his first stirrings when he has a dream about his friend, Fiona. His mother asks him to stay home a few minutes longer that morning than he normally would and she explains that he is ready to begin taking the pills that will stop the stirrings. Jonas was not very familiar with stirrings or what exactly they meant. All he knew is that they were to be reported immediately so people could begin "treatment." This indicates that the stirrings are seen as something "wrong" and treatable.

"He remembered that there was a reference to the Stirrings in the Book of Rules, though he didn't remember what it said" (Ch. 5).

All we know is that in his dream, Jonas and Fiona are in the House of the Old standing by a bathtub and he is bare chested and she is laughing at him. He is a little angry at her because "she isn't taking me seriously." He was apparently trying to convince her to take her clothes off and get into the bath tub so he could bathe her. The best way he can describe his feelings in this dream are "wanting." 

If the "Stirrings" are sexual desires, we can see why the community would want to keep these controlled. The Elders assign spouses and family units. Individuals do not get to choose their partners, and so there is no point in having sexual feelings which might cause them to be attracted to one another.

We also know that families to do not naturally procreate and have their own children. They are given children who are born from birthmothers, whom they never meet. Therefore we can assume that husbands and wives do not have sex and are not supposed to.

So, we can assume that the stirrings are closely regulated to prevent individuals from having sex with one another, choosing partners based on those "Stirrings," and procreating. All of those factors must be controlled by the community.