Yes, it's a reference to the emperor Nero and the famous story of how he played violin while Rome burned, suggestive of his cruelty and indifference to others' suffering.
It's part of that whole sequence of lines that place a positive against a negative to indicate how bad the "narrator" of the song is compared to the other, much better person, eg:
I know I’m weed killer, honey, and you’re sugar
If you’re the prosecution I get away with murder
We’re binary code, a one and a zero
You wanted violins and you got Nero
You wanted violins (love, romance) and you got a villain.
I think it's very clever.