Look at the temperature difference and thermal mass of the upper range of the radiative zone (~2,000,000 K) versus the temperature at the top of the convective zone (5,700 K). There would have to be a huge change in the temperature at the convective zone (at least a couple orders of magnitude) before you’d see any significant change at the RZ/CZ boundary that could feed back into the core, and such temperatures would destroy any conventional material long before they would alter the fusion dynamics of the Sun’s core.
Now, if you could surround the Sun with an opaque barrier right at the photosphere, that is a different point. There would be nowhere for the highly energetic charged particles and photons to go, and the conditions would change dramatically. But with 1.4044x10[sup]34[/sup] m[sup]3[/sup] to allow gas to expand into, it is going to be a very long time before you’ll get significant T=PV/nR such that it influences heat flux through the radiative zone.
I’m not certain the gas pressure or radiative load would be negligible.
Going with the Wikipedia solar wind numbers of 6.7e9 tons/hour with a velocity of 400 km/s. That gives a pressure of something like 0.003 N/m^2 at 1 AU (= 6.7e91e3/3600400e3/4/pi/150e6^2). That will equal a 0.006 G load when the Dyson sphere has a surface density of 0.04 kg/m2 (divide previous figure by 0.06 m/s^2). Picking a light mylar film at random, it has a surface density of 0.25 kg/m^2 (= 1.3928/1e3100^30.18e-3), which means the solar wind pressure would be about 1/6 that of the gravitational pressure.
So not negligible, although it shows the primary force will be in, not out.
I’ll let someone else do the radiative load. I expect it’ll be smaller than the solar wind load, but probable not many orders of magnitude smaller.
Edit: I just remembered that we’re assuming the particle flux is reflected back. That doubles the force I computed above.
Actually, the radiation pressure is significantly larger than the solar wind pressure. Still, though, that’s easy enough to deal with: You just make your shell the right thickness so that the weight and the radiative pressure balance each other out.
Remember that the pressure is (more or less) equal around the entire interior, so it isn’t as if we have to treat elements of the shell as being cantilevered or simply supported, nor will they experience a shear force (globally, though depending on the kind of support structure they may have local load peaks). As Chronos points out, this is easily mitigated by making the shell of a density such that gravity acts to counter the outward pressure. Of course, as pressure and temperature build up over eons, the forces will increase. However, the limitation will still be the temperature capability of any real-world material. Mylar[sup]TM[/sup] has a service temperature of about 150°C (423 K) though I doubt that it would stand up to the UV for very long, even if the interior were well-silvered. 6061 aluminum has operating temperatures in the 200°C to 250°C range. Once you get to an equilibrium temperature at this range (which is dramatically lower that what would be necessary to create a coupled thermal feedback to the core of the Sun) the shell is just going to come apart like wet toilet paper.