What would happen to a star if a Dyson sphere lined with mirrors reflected a significant portion of the stars light back to the star I have looked for similar questions here on stack exchange. The closest example to this that I found is Could a Dyson sphere destroy a star. That question assumed less than perfect absorption of material lining the inside of the sphere, and the effect of thermally re-radiating it inside the sphere.  A lot of the answers focused on that much of the thermal radiation would not be reflected back to the star, and is likely to be reabsorbed by the sphere at another location. On the other hand, this question asks what would happen if the Dyson sphere, using mirrors, deliberately refocus a significant portion of the rays directly back at the star.
 A: A partially reflective Dyson sphere is equivalent to asking what happens if we artificially increase the opacity of the photosphere - akin to covering the star with large starspots - because by reflecting energy back, you are limiting how much (net) flux can actually escape from the photosphere
The global effects, depend on the structure of a star and differ for one that is fully convective, or one like the Sun that has a radiative interior and a relatively thin convective envelope on top. The phenomenon could be treated in a similar way to the effects of large starspots. The canonical paper on this is by Spruit & Weiss (1986). They show that the effects have a short term character and then a long term nature. The division point is the thermal timescale of the convective envelope, which is of order $10^{5}$ years for the Sun.
On short timescales the nuclear luminosity of the Sun is unchanged, the stellar structure remains the same as does the surface temperature. As only a fraction of the flux from the the Sun ultimately gets into space, the net luminosity at infinity will be decreased. However things change if you leave the Dyson sphere in place for longer. 
On longer timescales, in a star like the Sun, the luminosity will tend to stay the same because the nuclear burning core is unaffected by what is going on in the thin convective envelope. However if a large fraction of the luminosity is being reflected back then to lose the same luminosity it turns out that the radius increases and the photosphere gets a little hotter. In this case, the radius squared times the photospheric temperature will increase to make sure that the luminosity observed beyond the Dyson sphere stays the same - i.e. by $R^2T^4(1 - \beta) = R_{\odot}^2 T_{\odot}^4$, where $\beta$ is the fraction of the solar luminosity reflected by the sphere. 
The calculations of Spruit et al. (1986) indicate that for $\beta=0.1$ the surface temperature increases by just 1.4% whilst the radius increases by 2%. Thus $R^2 T^4$ is increased by a factor 1.09. This is not quite $(1-\beta)^{-1}$ because the core temperature and luminosity do drop slightly in response to the increased radius.
It is probably not appropriate to quantitatively extrapolate the Spruit treatment for very large values of $\beta$, but why would you build a Dyson sphere that was highly reflective? Qualitatively, the envelope of the star would expand massively in response to the heat being deposited in it from outside and in this case the photosphere might become cooler, despite the extra heat inflow.
The above discussion is true for the Sun because it has a very thin convection zone and the conditions in the core are not very affected by conditions at the surface. As the convection zone thickens (for example in a main sequence star of lower mass), the response is different. The increase in radius becomes more pronounced; to maintain hydrostatic equilibrium the core temperature decreases and hence so does the nuclear energy generation. The luminosity of the star falls and the surface temperature stays roughly the same. 
A: An analysis would have to look at the effects over time. With my limited understanding of physics and intuition, I see the outer layers of the star reabsorbing the rays. Where the outer portions of the star until now have experienced large amounts of energy flowing in one direction, now has a net outward energy flow of maybe 1/2 to 1/10 of what it used to. 
Initially, I see the inner core should continue it's fission process as normal. I see the outer layers absorbing much of the extra energy and begines to expand outward.  
Assuming so far I am on the right track, and say at some point the star has grown to fill 1/2 of the volume of the Dyson sphere, much of the matter is significantly displaced. Though the center of gravity has not changed, if I am correct, there should be less pressure on the core and it should have shrunk, and thus the rate of fusion has slowed down and the radiation emitted from the surface of the star should now be much cooler than when the mirrors first began to redirect the light back to the star. Furthermore, the star's rotation would have slowed down; solar flares would be smaller or none existent. The last statement may not be true for awhile until the core's spin also slows down.  
Ultimately, If i have not made an error in my general assumptions, maybe it could be possible to expand and cool say our sun to maybe fill a Dyson sphere the size of of a Mercury or Venus orbital diameter. That reduced fusion rate in the sun would reduced the habitable zone radius, making a habitable small Dyson sphere more easily realizable for an advanced civilization. 
a little off topic but heat pipes with turbines in them could have their evaporators on the inside the Dyson sphere and their condensers on the outside. Concentrating the thermal energy though heat pipes would give an advanced civilization a way to collect thermal energy and allow them to build a habitable Dyson sphere even smaller than what is typically considered the habitable zone around a star. 
Seeking confirmation, corrections, comments, and or other possible answers. 
A: In Orion's Arm, this is called a Starbooster, and it has the effect of heating up the star's outer layers, causing its spectrum to shift upwards. It's typically used to change an M- or K-class star into a Sun-like G-class.
That said, Orion's Arm is science fiction (albeit science fiction that purports to be as realistic as possible), so I'm not sure how realistic this is.
