That is not how time dilation works. It is not a 1 to 1 rate or people who are twice as heavy would experience them moving at super speed. At the surface of the Sun, time dilation is at a rate of ~67 seconds per year. Or, for every year on Earth only 364 days, 23 hours, 58 minutes, and 53 seconds will have passed.

As for how the star runs out, the end result of fusion is that two hydrogen atoms turn into 1 helium atom. Eventually, all the hydrogen atoms in the star have been fused. Of course there will be some hydrogen left in the star, but it is too diffuse to continue fusing at a rate to sustain the star. Remember that stars are a delicate balance between the pressure exerted by all the heat and fusion, and the gravity trying to contract it.

Last, the sun is 1.3 million times the size of Earth by volume and 333,000 times the mass of Earth.