For a launch, I'd say no. About the only scenario where it could be bad, at the planetary level, would be if the mission was to go out, say past pluto, and redirect a comet. And something goes wrong, and you end up aiming the comet at some planet.
Even that would take a lot. But if we posit that the original mission plan was to move the comet. Just that the execution screwed up, and moved it the wrong way -- then maybe.
That still wouldn't affect the whole solar system. It would ruin your day if it hit the planet you were on. But the rest of the planets would be fine.
unless you want to talk about timescales of billions of years. Then.... maybe ... just possibly... you could get a butterfly effect. where your ship slightly changes the orbit of hundreds or thousands of pluto-distance comets and over time, we get unlucky enough that several planets get impacted, maybe several times.
so, maybe earth (well - Life on Earth - the rock itself wouldn't notice), the moon, maybe some bases on a few other moons, would be toast?
as far as hurting a gas giant, or the sun. NO. Not Gonna Happen. There's a XKCD "What if?" comic about "what are the chances of accidentally affecting Jupiter's orbit?" Suffice to say, you cannot accidentally affect a gas giant in any meaningful way.