How would fuel consumption work at the speed of light? (hypothetically)?

As you gain higher velocities your mass also increases so the amount of fuel required increases with the mass you have gained. To be honest I don't want to do any actual calculations except to say the amount of fuel you need for the journey isn't a significant % of the total fuel you will need. The vast majority of the fuel you will need will be burnt off just getting your ship to accelerate to those speeds, once at those speeds you will not need to burn much fuel at all during that journey, then you will need to burn off almost the same amount of fuel again in order to decelerate, so your question about the time dialation effect seems to be mute.