How fast is too fast to stay in orbit around any particular large body in the Solar System?
If we interpret the question as asking for the highest speed relative to the primary of any closed orbit, then it becomes answerable.
The orbit you want will basically skim the surface of the body (or the top of its atmosphere) at closest approach, and recede as far as possible before the perturbing effects of other bodies mean it would no longer be a closed orbit. For most bodies, the speed at closest approach will be just very slightly below escape velocity from that altitude. For a few, like the Gallilean satellites of Jupiter, it will be significantly less because perturbing effects from the nearby large body (ie Jupiter) will be large.
The highest value in the solar system will certainly come from the Sun, at over 600 km/s. The next highest is Jupiter at around 60 km/s.
What you are looking for is called the escape velocity. That is the velocity which, once reached, will enable you to leave orbit around the object. It is a function of the mass of the object around which you are orbiting, and the distance you are away from said object. The formula is as follows:
$G$ is the gravitational constant, $M$ is the mass of the object, and R is the distance away from it you are. Note that in most instances we know the product of $G*M$ better then either $G$ or $M$, so it is often easiest to use those. Wikipedia has a good list of the escape velocity of various situations in the Solar System.
Surely it depends entirely on the mass of the object, and its centre of orbit, because those two factors would determine the velocity. Plus the higher the velocity the further from the centre of the solar system surely. Thus it would be the velocity required for the mass to reach the edge of the heliosphere.