Therefore I was thinking: how does Typst want fonts to be formatted? Because It’s been very hit and miss with a lot of these color emojis to get them in SVG.
Could you give a screenshot of your result? Are you using the font shipped on typst.app, or compiling locally with a font that you have downloaded on your own? Are you exporting to SVG, PNG, or PDF?
Noto Color Emoji should just work with Typst v0.14.2. It was once broken, but it has been fixed since v0.14.0-rc.1.
#set text(5em, font: ("Libertinus Serif", "Noto Color Emoji"))
#"\u{1f4d6}"
Noto Color Emoji provides a bitmap font file and a vector font file. Even if you compile with Typst to SVG, depending on your local font installation it’s possible that Typst will use the bitmap font. And the bitmap emojis will render correctly so maybe that’s what @Y.D.X is seeing, and you’re probably using the vector font which has more issues both on the encoding (Typst) side and rendering (viewer) side, see for example this issue.
In any case as @Y.D.X says it would be helpful to have more details on your setup and how you compile.