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VS Code has support for it on a per-theme basis. I am currently using the Atom One Dark theme on VS Code, and Operator Mono, it does work:

https://puu.sh/sbUti/6022c0afb0.png

Above being a random screenshot from a project laying around.

I haven't found how to persist those italic settings. I can use the inspector and set them, but as soon as I close the app I loose them.

As I mentioned, it's on a per-theme basis.

Try installing 10-15 themes, and then trying them one by one. Some will have italic comments, some won't.

I use Source Insight at work for C code and it can highlight syntax further than many editors I've used. You can specify bold, italic, font, size, colors, for every different token.

I think it's Windows only though.

Terminal vim handles italics, assuming the terminal does.

Right, but what terminals do that?

I use mlterm, which aside from italics and 24-bit-color support, has the additional advantage for vim that you can configure the Alt-key prefix away from ESC and have a longer timeout.