Disallowing \textbf, \it, \sffamily, ... in argument of macro or environment

There's no list of the font changing commands available, but you can redefine \selectfont, inside a group, to do nothing except issuing an error message.

\documentclass{article}

\makeatletter
\newcommand{\restricted}[1]{%
  \begingroup
  \let\selectfont\cgogolin@error
  #1%
  \endgroup
}
\newcommand{\cgogolin@error}{%
  \@latex@error{Font change not allowed}
    {You used a font changing command, which is\MessageBreak
     not permitted here}%
}
\makeatother

\begin{document}

\restricted{calling me {\bfseries like} this \textit{is} not allowed}

\end{document}

Here's the output on the terminal:

! LaTeX Error: Font change not allowed.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              

l.19 ...series like} this \textit{is} not allowed}

The following minimal example provides a selective font switch check by using etoolbox's \patchcmd. Each font switch is used to patch the argument of \restricted. If it is successful, an error is generated, otherwise nothing happens.

\documentclass{article}

\usepackage{etoolbox}

\makeatletter
\newcommand{\restricted}[1]{%
  \def\restricted@arg{#1}%
  \renewcommand{\do}[1]{%
    \patchcmd{\restricted@arg}% <cmd>
      {##1}{##1}% <search><replace>
      {\cgogolin@error}{}% <success><failure>
  }
  \docsvlist{\bfseries,\textit}% Check for certain font uses... add more
  % <do stuff with #1>
}
\newcommand{\cgogolin@error}{%
  \@latex@error{Font change not allowed}
    {You used a font changing command, which is\MessageBreak
     not permitted here}%
}
\makeatother

\begin{document}

Calling me {\bfseries like} this \textit{is} allowed.

\restricted{Calling me {\bfseries like} this \textit{is} not allowed.}

\end{document}

This comes with the caveat that it tests for explicit use of any particular font switch. For example, \bfseries my be allowed if only checking for \textbf, say. Also, no expansion is performed on the argument, so hidden font switches won't be picked up (for example, if \let\mybfseries\bfseries is used).