I've watched a few theoretical videos about Dyson spheres where they mention that the most practical way to do that would be to disassemble a planet - preferably one already close to the sun - Mercury or Venus.
Question: Would the solar system become completely unstable if one (or both) of those planets got disassembled? As a follow up question: what if one or more of the gas giants somehow were taken out of the solar system?
Not really. The mass of Mercury and Venus together are just 0.0027 of Jupiter's mass. So the effect is really small.
Now, in principle, there are resonances where the regular tug from planets affect the long-term evolution of the orbits of other planets. The exact details get very messy, but in some cases the presence of a planet stabilizes certain orbits - or destabilises them (this is very visible in the asteroid belt). But the planetary orbits look largely non-resonant. So if two planets went missing there would not be much of an effect.
It is likely that this is true for Jupiter too. What would happen is that the Trojan and Greek asteroids would drift out of their 1:1 resonance and end up in a loose belt, and the asteroid belt would also reorganise.
Overall, you don't need much mass for a Dyson sphere if you want to just collect energy. My paper on using one to settle the universe disassembled Mercury mostly as a conservative estimate to get more than enough material - I currently think you need a few largish asteroid masses, maybe Ceres (assuming it has the right elements, an important if). Conversely, Dyson element orbits will be subject to gravitational interactions from remaining planets that will cause precession and other possibly unwanted shifts one need to compensate for. But a Dyson sphere is mass-wise a microscopic thing compared to planets.
One of the possibilities for the inner Solar System becoming unstable over the next few billion years is a secular resonance between Mercury and Jupiter that increases the eccentricity of Mercury's orbit, which can result in some interesting consequences. From Batygin & Laughlin (2008):
The experiments yielded one evolution in which Mercury falls onto the Sun at ~1.261 Gyr from now, and another in which Mercury and Venus collide in ~862 Myr. In the latter solution, as a result of Mercury’s unstable behavior, Mars was ejected from the Solar System at 822 Myr.
If Mercury and Venus were not present, these outcomes would presumably not happen (although it's not clear that either of the scenarios mentioned above will actually happen in reality). It's not clear whether or not other instabilities might show up before the Sun becomes a red giant though.
I feel like this is getting into worldbuilding a little, but a Dyson sphere is problematic. Dyson ring or swarm is more possible.
One change, regarding the removal and redistribution of mass of Venus would be a change in Earth's eccentricity cycle, as Venus is one of the key planets that effects Earth's eccentricity, along with Jupiter and Saturn. That could lead to a change in Earth's ice age cycle, which is currently on a roughly 100,000 year pattern which corresponds to one of Earth's eccentricity cycles. Granted, any species advanced enough to disassemble a planet would be able to control climate cycles.
