Ignoring the impossibility of the sun suddenly collapsing, and the energy release (which would kill us): If the sun suddenly turned into a black hole, could we survive if we had sufficient energy for heat, light, etc?
I assume that what you basically mean is that
The answer is NO. We would just be orbiting it as we do now. There is no reason why should there be any effect on gravitational force on Earth from the Sun, since it is dependent on total mass and distance from Sun's center to Earth's center. None of these factors change.
If you're still confused, pay attention to the fact that there is a supermassive(about a million times the Sun) black hole at the center of our Galaxy and the whole Galaxy is orbiting it in a way, nothing really happens to it.
Always remember that black holes are not something which sucks everything around it. If that would have been the case, the universe would be far more empty. They have got a bad impression just because they don't let light escape, not because they suck everything around them.
I wonder just for few day's till our resource are available for food and heat. But it will not change because it's mass is not sufficient to become a black hole