Assuming you knew how to do the plumbing -- I'm sure it would require massive quantities of handwavium, at the very least -- you could use the Sun's energy output to move the Sun. But how fast?
The mass of the Sun is 2x1030 Kg, the power output of the Sun is 4x1026 watts. Assume that you have a photon drive so that the momentum flux is power/speed of light. (This is the most efficient possible reaction engine.) Also assume no losses.
With a photon drive, you get A=P/cM where A is the acceleration, P is the power (energy/second), c is the speed of light, and M is the mass.
Plugging in the Sun's numbers, its acceleration turns out to be 7x10-12 meters/sec2. After a year under power the Sun would be blistering along at .0002 meters/sec.
(This is not to say that given enough time you couldn't do amazing things using this drive. In a million years the Sun would be moving 200 meters/sec and in 100,000,000 years it would be moving 20 kps and in a half-billion years it would be far outside the Galaxy. But on a human scale, not so much.)
The reason is that while the Sun puts out a terrific amount of energy, it's also very, very, very massive. A good way to think about it: In about ten billion years it fuses all its hydrogen to helium. That's actually a very slow rate -- compare with a conventional rocket which burns 80% of its mass in ten minutes -- and burning its fuel so slowly means also that its acceleration is not exactly peppy.