No. Consider a small perturbation (i.e. Poking) the ball on your table. At least for a small enough poke and/or a big enough table, friction slows down the ball again, and it stops.
Now consider a small poke on a Dyson sphere that was (before the poke) perfectly balanced. The part that got poked is slightly closer to the sun, and the point opposite is slightly further away. So the closer part has some gravitational force acting on it, and the opposite part has a lower gravitational force acting on it: O(1/(r-e)^2) vs O(1/(r+e)^2). The imbalance causes a net gravitational force in the direction of the poke, which eventually crashes into the sun.
No, it's neutral--the equivalent of keeping a ball on a table.