There will be no change observable by naked eye from any planet in the Solar System.
It's easy to estimate the distances for which change is visible. There are two key points. First, changes in the constellations are due to parallax. Second, by definition, a baseline of 2 AU (astronomical units) produces a parallax of 1 second of arc at a parsec's distance (wihc turns out to be about 3 light years) and half an arc second at two parsecs, and so forth. Double the displacement to four AU and you double the parallax.
The nearest visible star to the Sun (Alpha Centauri) is just over a parsec away. For it to move a barely noticeable distance (say a tenth of a degree or 1/5 the diameter of the Moon) in the night sky, the baseline would have to be 360 times as long, since 0.1 degree is 360 arc seconds. So the closest star in the night sky would have moved a just-perceptible amount if observed from 500 AU away from Earth.
The brighter stars which make up the constellations range from the distance of Alpha Centauri out to thousands of light years, but many of them are within 100 light years (30 parsecs). While a movement of 0.1 degree is perceptible, for the constellations to be obviously distorted, you'd probably want many stars to move by more than a degree (two full Moons).
So: 30 parsecs and 1 degree requires a movement of 1AU * 30 * 3600 (arc second/degree) or about 100,000 AU. That's about 1.5 light years.
If you voyaged away from the Sun, a keen observer would notice an slight effect somewhere out past the Kuiper Belt, and a few stars would be massively displaced by the time you were out in the Oort cloud, but for wholesale disruption of the constellations, you'd have to go about a light year.
Two points: First, at a light year's distance from the Sun, the constellations would still be somewhat recognizable, though there'd be no doubt at all that things were scrambled.
Secondly, the shifts would be largest in the direction perpendicular to your line of travel (the baseline for parallax) and much smaller fore and aft.