It gets dark. Seriously, that’s a big problem. We can’t live without sunlight at present, even with technology. No sunlight = no food.
Eating the sun doesn’t necessarily mean eating the Earth. We’d still be in orbit around the hole after the sun was gone.
However – and this was answered better by Ollie earlier in the event – to eat the Sun, the hole has to get to the Sun; it can’t do it at a distance. That means we have something at least several times as heavy as the Sun floating through the solar system. So the first problem is that it scatters comets all over the system as it comes in through the Oort Cloud, and comets hitting planets make very big, very deadly explosions. The second problem is that the planetary orbits are disrupted, so Earth might even get thrown out of the system into the dark.
The other “little” problem is that as the Sun’s gas gets sucked into the hole it heats up and emits deadly radiation. Therefore, if a black hole does come to tea in our system I’m afraid we won’t survive the visit even for a little while.