No, you do not need a special tire for indoor bike trainers. However, it can be a good idea to invest in a special bike trainer tire. There are two reasons for this. Firstly, a specialty tire will be less noisy than a conventional tire. This will be the case whether you are using wind, magnetic or fluid trainers. Specialty bike tires are also more durable than conventional tires. That is because they are constructed from a special rubber blend.
Of course, the hassle of switching your tire over every time you use your indoor cycle trainer needs to be weighed against these two benefits.