If a trash can of nuclear waste is motionless relative to the sun, it would be simple to push it toward the sun. The same would be true of a trash can motionless relative to a black hole.
But any trash can on earth is not motionless relative to the sun; it is orbiting the sun at 67,000 mph. Imagine you were at the end of a rope and I was swinging you around at 67,000 mph. Think of how hard it would be to climb up that rope to me.
In order for the trash can to become motionless relative to the sun, we would have to launch it in the opposite direction of the earth's orbit at 67,000 mph. It is this amount of energy needed that people refer to when they say it is not worth sending the waste to the sun.
On a related topic, this same principle is one of the reasons we tend to launch missions to planets further from the sun than us (Mars, Jupiter, etc.) rather than those closer to the sun than us (Mercury, Venus)