Admittedly, I'm only 60 something episodes in, so this may already have an answer. However, as hypothetical as this may be, due to the fact it seems impossible, I'm simply curious what hardcore Fairy Tail fans think might happen if Natsu were able to eat the fire of a star? Just how strong might he become? How much might his physical form change? Would a star's gaseous fire even be compatible? If you've got the knowledge, I wouldn't mind someone applying some "scifi" theoretical science and possibly even math to the answer. Looking for a great read of an answer! Good luck and thank you!
TL:DR, Natsu would be an OP fire god, and would never need to eat fire ever again
So as far as we know, Natsu's magic allows him to absorb fire (whether magical or natural) and lets him redirect it and use it however he wants. From all references there is no canon drawbacks from consuming exuberant amounts of fire, nor do we know the absolute limit of how much fire Natsu can consume. So within the Canon of Fairy tail, if Natsu actually consumes a star, he might as well not need to ever eat any more fire, unless it was in defense; he would be able to spout fire and use fire magic at any time.
However, let be speculative for a minute and try to SCIENCE this out. If Natsu was able to consume any amount of fire, dragon killer magic has to be able to preserve ones internal organs from being ripped apart by the heat and energy generated by any form of fire. But a star is a unending fusion reaction, and the mass of a star is capable of affecting large masses of matter. Thus if Natsu was able to consume a star, he would be consuming all of that mass. Given that Natsu's canon height of 175cm, and the average human weight of 65.3 to 79.8 kg, with an potential volume of around 0.06 to 0.08 m^3. If Natus was to consume the smallest possible star (a red dwarf, at the minimum weighs about 7.5% of our sun) would be gaining 1.49 x 10^29 kilograms. This will not generate a black hole, but hypothetically, this much mass will effectively cause the planet to envelop Natsu at is core. Not only killing him, but reshaping the planet, and killing everyone on it. So Natsu would be a fire god, and would never need to eat fire again, but he would also be the new core of the planet.