Earth would not remain flat for a long time as gravity will make sure it again pops up into a sphere. Remember, gravity pulls equally from all sides, hence planets are spherical.
Now, if earth was flat and not a sphere, then it would obviously stop rotating. This will lead to the following problems.
Firstly, only revolution around the Sun but no rotation around its axis would mean that Earth will experience a 6-month long day, followed by a 6-month long night.
Secondly, it is believed that rotation of Earth generates its magnetic field. Without rotation, there would be no magnetic field to protect us from the harmful solar winds. Also, it would mean an end to the world-famous lights called Auroras as they are formed due to the interaction between Earth's magnetic field and particles of solar wind.