What would happen to the Sun if you reflect all its emitted e.m. radiation back? What would happen to the Sun if you would reflect, in whatever way, all the outgoing electromagnetic radiation (Solar winds can be neglected)?
 A: This is the same as asking what would happen if the sun couldn't get rid of the heat it generates. In that case, the heat builds up, the sun's temperature goes up, and in response the sun expands a bit. This expansion causes the fusion reactions in the sun's core to slow down, which slows down the rate of heat generation. But if none of that heat can escape, the sun will get hotter and it will continue to expand, and the fusion rate in its core will continue to decrease. At some point in this process the sun goes blown up so much in size and the temperature in its core falls so far that fusion can no longer occur and the sun's power source is shut down.
A: Some facts (see also the picture below):
The circular core where fusion (the source of the emergent energy) takes place has a radius of about 175 000 kilometers. The radius of the whole Sun is about 700 000 kilometers. Because of the enormous pressure, the plasma density in the core is about 150 times the density of water. The temperature here is 15 000 000 Kelvin.
It takes tens of thousands of years for energy produced in the core to reach the photosphere, the outside of the Sun which we can see, and which has a depth of only 1000 kilometers. The temperature inside the photosphere is about 5700 Kelvin. The density here is about $3\times10^{-4}(\frac{kg}{m^3})$.
In the radiative zone, the density drops from $2\times 10^4 \frac{kg}{m^3}$ (about the density of gold) down to only $20(\frac{kg}{m^3})$ [less than the density of water, about $10^3 (\frac{kg}{m^3})$] from the bottom to the top of the radiative zone. The temperature falls from 7,000,000° Kelvin to about 2,000,000° Kelvin over the same distance, about 300 000 kilometers.
The remaining 225 000 km consists mainly out of the convective zone. Convective motions carry heat quite rapidly to the surface. The fluid expands and cools as it rises. At the visible surface, the temperature has dropped to 5,700 Kelvin and the density is only $2\frac{kg}{m^3}$ (about 1/10,000th the density of air at sea level).
Now, the total energy contained in the Sun, $E_{sun}=m_{sun}c^2=1,74\times 10^{47}(J)$. Its binding energy can be neglected. The total fusion potential is about $1,2\times 10^{44}(J)$, so it's safe to say that the Sun's mass stays the same during its lifetime.
Now, considering these facts, what would happen if the energy radiated away would stay inside the Sun? Picture it. The photosphere and convective zone would heat up, which makes these layers expand, obviously. Too tittle though to have any influence on the core's density. The non-radiated heat will reach the core after tens of thousands of years, which is practically in zero time compared to the Sun's lifetime. After the core has absorbed the energy it will heat up, but its temperature will rise insignificantly.
To sum it up: In practice, we can see no difference between the time fusion takes when the radiating heat would be reflected back, so nothing really changes though. When fusion processes have died out, the Sun will follow the typical scenario of a star with the Sun's mass.

