How do I get italic sans-serif in math mode?

Math alphabets commands such as \mathit and \mathsf are not “cumulative”. You have to allocate a specific math alphabet.

\documentclass{article}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{bm}
\usepackage{fix-cm} % not needed if not using Computer Modern

\DeclareMathAlphabet{\mathsfit}{T1}{\sfdefault}{\mddefault}{\sldefault}
\SetMathAlphabet{\mathsfit}{bold}{T1}{\sfdefault}{\bfdefault}{\sldefault}

\begin{document}

$C$ default

$\mathit{C}$ mathit

$\mathcal{C}$ mathcal

$\mathsf{C}$ mathsf

$\mathsfit{C}$ mathsfit

$\bm{C}$ bm default

$\bm{\mathcal{C}}$ bm mathcal

$\bm{\mathsf{C}}$ bm mathsf

$\bm{\mathsfit{C}}$ bm mathsfit

\end{document}

Depending whether you are only seeking text characters, you could use the math \text command:

  \[ \text{\sffamily\itshape  ABC abc}\]