A guy I know was a sports talk host at his station for many years. When a better time slot opened at a cross-town station, he mentioned to his producer and board op that he had applied. Soon after, his program director heard the news. Within hours, the host was fired.
Looking for your next job while currently employed is not uncommon. The best time to look for a job is when you already have one. However, there are certainly smart ways to do it.