No Plate Tectonics
To have a world with no mountains occur naturally, there must be no plate tectonics. To have no plate tectonics, there must either be no moon, or the moon must be tidally locked to prevent its gravitic influence causing heat-producing deformation of the world.
A consequence of this is that there would be no earthquakes and erosion would eventually level land and sea-floor alike until it was eventually all an equal depth beneath the water (aside from small solar tides). There would be no geomagnetic field (that being caused by the same processes as plate tectonics), which would allow higher radiation levels at the surface.
The Mountains were taken away!
If earth's mountains were magically instantly removed, there would be immediate repercussions. All that stone has mass, and the crust beneath which had been pressed into the mantle would rebound, causing worldwide earthquakes. Volcanoes would erupt as their plugs of stone were removed or weakened.
All this is not considering the effects on the global weather patterns.
In the long term, mountains would reappear as plate tectonics continued to deform the landscape.