Not a couple of tons, thousands and thousands of tons. A rocket to move any significant mass of waste to the sun (far more delta-v than getting to the moon, or escaping the solar system), without decades-long journeys [1] would make Saturn V and Starship look like toys, or if would be a very complex asssemble-in-orbit deal.
[1] you can use gravity assists like the Parker Solar Probe, or go very, very far out, make a small adjustment and then fall back in for about 1/3 the total delta-v, which is still a rather large amount.
[1] you can use gravity assists like the Parker Solar Probe, or go very, very far out, make a small adjustment and then fall back in for about 1/3 the total delta-v, which is still a rather large amount.