Is the flat area in just one spot on the tire? Indoor trainers are hard on tires....... If you ride the trainer frequently, the tire will wear down in the middle and it will be flat around the entire tire. I usually put an old tire on my trainer bike so I don't have to replace a good one at the end of winter.
If you only ride the trainer occasionally, it shouldn't be a problem.
Time is a companion that goes with us on a journey. It reminds us to cherish each moment, because it will never come again. What we leave behind is not as important as how we have lived." Captain Jean Luc Picard