The answer to this question is quite simple, it depends on the object. See, if an object is already moving, it will continue to move even if no Net Force is applied. So by this way, yes, the object can move or keep moving without any application of Net Force. But if an object is not moving and is at rest, then it will be stationary if no Net Force is applied. To put it simply, the object will continue doing what it was doing initially if no Net Force is applied.