Ummm, sorry about this people....
The sun ISNT burning anything!
It is the process of
fusion of Hydrogen atoms into Helium which is causing the tremendous outpouring of heat and light within the vicinity of our Solar System....
"Burning" implies a fuel combusting in oxygen which is consumed to release the energy contained within the fuel.
The sun is not doing this. There is no Oxygen in space and the "burning" of the sun is simply the appearance of the release of the energy which sort of kinda "looks" like its on fire and therefore with us feeling the heat, we relate it to fire and burning....
The mass of the sun and the extreme density and pressure of the Hydrogen contained within it is causing the fusion. The mass of the sun causes the gravity which keeps the orbits of the planets and our solar system in check.
Due to the extremely high gravitational attraction that a black hole would impart on our sun if it were to come within range, the mass of our sun would begin to peel away and start to fall within the black hole's influence, effectively reducing the sun's mass and therefore EATING the sun rather than burning it. Once enough mass is stripped away from the sun, it would extinguish as the balance of mass/pressure/fuel would not allow fusion to continue....
That's my answer....and I'm sticking to it....
Cheers
Chris