Citing the manual, chapter 2.1:

“Note that many of the examples use fonts that are not included in TEX Live or
MiKTeX, and some of them are non-free fonts that must be purchased.
I’d like to reduce the number of non-free fonts used in this manual. If you know
any freely available fonts that could be used as alternative to any of the fonts in
this document, please suggest them to me.”

In this sense and following a discussion on de.comp.text.tex I started to put together a list with all the non-free fonts used in examples and try to find replacements. IMO the fonts should come from the TL-collection, also I try to use as many different fonts as possible as this is somehow a showcase situation. At the moment I don’t consider AAT fonts (not having access to a Mac), AAT examples are marked with *, AAT fonts with † and AAT fonts that are only used in examples of AAT features are marked with ‡. In cases where the examples are font independent the proposal is marked with a pilcrow ¶.

Non-free fonts

• Georgia [Scale/MatchLowercase] → ¶LibreBaskerville with
  Verdana → ¶Cabin or ¶DejaVuSans,
  [MatchUppercase] with Arial → ¶PT Sans;
  (use fonts with clearly visible different vertical metrics)

• Helvetica Neue (Ultralight+Regular) → ¶SourceSansPro (ExtraLight+Regular) or ¶Raleway (ExtraLight+Regular); (use fonts that come with extra thin weights)

• Cochin [Mapping feature] → ¶fbb or ¶AntykwaTorunska;

• Didot [LetterSpace] → ¶GFSBodoni;

• Hiragino Mincho Pro [Style: ital ruby hkna vkna] →
  [Annotation: nalt] →
  [Shape: trad (smpl jp78 jp83 jp90) expt nlck; Char width: pwid fwid hwid *twid
  *qwid (palt halt)]

• Adobe Jenson Pro [hist] →

• Adobe Garamond Pro [titl] → (perhaps EBGaramond once implemented)

• Warnock Pro [sups, numr, dnom, sinf, ordn] → LibreCaslon,
  [cswh] → (EBGaramond12-Italic, as soon as implemented)
  [salt] → Quattrocento (change example string to "M Q W");

• † Hoefler [Defining a single font face] → virtually any font but needs
  different features in the example;
  ["features for, say, just italics"] → ditto, something like

  \fontspec{EB Garamond} \itshape Don’t Ask Victoria! \\
   \addfontfeature{ItalicFeatures={Style=Swash, Ligatures=Historic}}
  Don’t Ask Victoria! \\
  

  *[Defining new features];

• ‡ Skia *[Multiple Master–like features in AAT fonts];

• ‡ Hei Regular

• ‡ Zapfino *[Defining new features]

Free fonts
For completeness here are the free fonts that are already in use in the manual (in the order how I found them):
TG Pagella (body), TG Bonum, LM Sans, Inconsolata, Kurier, TG Adventor, TG Termes, TG Chorus, TG Schola, Linux Libertine, LM Roman, Persian Modern, Junicode, EB Garamond, Romande ADF, Charis SIL, Doulos SIL