Not always. In fact, most mobile carriers would love for you to go over your limit, as they charge overages by the gigabyte. Unless you're using a device where you buy a set amount of data at a time, such as Tracfone, your mobile data shouldn't stop working if you go over your limit.
Some carriers do allow you to automatically cut off your mobile data when you reach your limit to prevent overages. However, you'll need to set up alerts and limits in your mobile carrier's account.