It would also be far cheaper to send that nuclear waste to Mars. It takes 55 times the energy to fly to the Sun than it does to go to Mars:
https://www.nasa.gov/solar-system/its-surprisingly-hard-to-g...
Sure but since we're talking about pie in the sky stuff requiring tech we don't yet have to begin with, putting it into the sun is a better permanent solution.
I don't think Elon is ever going to colonize Mars, but other people may someday.
In addition, even sending it to Mars would be extremely expensive.
Shower thought: Why send it to Mars when we can send it to uninhabitable Venus? Even gravity might work in our favor.