How do I change the color of comments in VS Code?

From 1.15 (July 2017) you can change it from settings.json Ctrl+,

"editor.tokenColorCustomizations": {
    "comments": "#d4922f"
},

From 1.20 (January 2018) you can also do it for each theme separately:

"editor.tokenColorCustomizations": {
    "[Atom One Dark]": {
        "comments": "#d4922f"
    }
},

Or now you can specify settings for multiple themes at once as "[Atom One Dark][Tomorrow Night Blue]": {...}

Finding the right scope:

To find the right scope, you need to open up the tokens and scopes inspector. Press Ctrl + Shift + P to open up the command palette, and then type and search for Developer: Inspect Editor Tokens and Scopes, as shown here:

Select it. Now, you can click around on different syntax to see the syntax highlighting scope and information, like this:

When done, hit Esc to exit the tokens and scopes inspection mode.

Selector priority:

https://code.visualstudio.com/blogs/2017/02/08/syntax-highlighting-optimizations#_textmate-themes

Ok, more examples (for js):

"editor.tokenColorCustomizations": {
    "textMateRules": [{
        "scope": "INSERT_SCOPE_HERE",
        "settings": {
            "foreground": "#ff0000"
        }
    }]
}

comment
punctuation.definition.comment
comment.block.documentation
storage.type.class.jsdoc
entity.name.type.instance.jsdoc
variable.other.jsdoc