How do I change the math italic font in XeTeX/fontspec?

If you do not wish to otherwise change your maths symbols, the best solution is to use the mathspec package:

\documentclass{article}
\usepackage{mathspec}
\setallmainfonts(Digits,Latin){Georgia}
\begin{document}
Hello $a+\mathrm{b}=c$
\end{document}

Here, Georgia will be used for the body text and \mathrm, and Georgia Italic will be used for the italic math glyphs.

The unicode-math package (which I kinda wrote) can also do this, but it's somewhat overkill for your purposes and has the additional downside that you need to also load a Unicode mathematics font. Here's an example:

\documentclass{article}
\usepackage{unicode-math}
\setmainfont{Georgia}
\setmathfont{xits-math.otf}
\setmathfont[range=\mathit]{Georgia Italic}
\begin{document}
Hello $a+b=c$
\end{document}

Note that mathspec will not run (yet) on LuaTeX, so if you need a LuaLaTeX solution then you'll need to use unicode-math for now.

Thanks to Will and Leo for pointing out the mathspec package. So now I use:

\usepackage{mathspec} %loads fontspec as well
\setmainfont{Gotham Book}
\setmathrm{Gotham Book}
\setmathfont(Digits,Latin){Gotham Book}

This snippet may need optimization, but works.

The Literal Answer

As of 2019, fontspec should find the bold and italic fonts, and load \mathrm, \mathbf and \mathit when you \setmathrm.

If you need additional set-up for \mathit, the ItalicFont and ItalicFeatures options of \setmathrm set up \mathit, just as the Boldfont and BoldFeatures options set up \mathbf.

\documentclass{article}
\usepackage[svgnames]{xcolor}
\usepackage{fontspec}
\usepackage[paperwidth=10cm]{geometry}

\setmainfont{TeX Gyre Schola}
\setmathrm{TeX Gyre Schola}[
  ItalicFeatures = {Color = green}]

\begin{document}
Text \textit{italic} \(\mathrm{math}\) \(\mathit{italic}\)
\end{document}

What You Also Might Have Meant

In practice, someone asking about how to set the \mathit alphabet likely means the alphabet used for $x$.

To set that, you want to load the unicode-math package, set a math font that matches your main font and then, if necessary, add a line such as

\setmathfont[range=it]{Gotham Book Italic}

You can do likewise for up, bfup, bfit and the other math alphabets.

In unicode-math, you would normally use \symit, \symup, etc. for individual letters used as variables, and \mathit, \mathrm, \mathup, etc. for complete words. However, you can give a package option to interpret \mathit as \symit and make migrated code compile correctly.