(Re)Define a character within newcommand

I'd define \schwa with a conditional:

\documentclass[10pt]{article}
\usepackage{fontspec}

\newcommand{\transcr}[1]{%
  \textit{\transcrtrue #1}%
}

\newif\iftranscr % starts out false
\DeclareRobustCommand{\schwa}{%
  \iftranscr\textsuperscript{ə}\else ə\fi
}

\begin{document}
In normal context, \schwa\ appears as it should.

In a different place it needs to be \transcr{sup\schwa rscript}, but it isn't

And then it doesn't appear as it should in normal context anymore: \schwa.

\end{document}

Since \transcrtrue is issued in the group provided by \textit, its effect is confined to it.

A slightly different approach allows for directly using ə (but also \schwa, if you prefer):

\documentclass[10pt]{article}
\usepackage{fontspec}
\usepackage{newunicodechar}

\newcommand{\transcr}[1]{%
  \textit{\transcrtrue #1}%
}

\newif\iftranscr % starts out false
\newunicodechar{ə}{%
  \iftranscr\textsuperscript{ə}\else ə\fi
}
\newcommand{\schwa}{ə} % must be done _after_ \newunicodechar{ə}{...}

\begin{document}
In normal context, ə appears as it should.

In a different place it needs to be \transcr{supərscript}, but it isn't

And then it doesn't appear as it should in normal context anymore: ə.

\bigskip

In normal context, \schwa\ appears as it should.

In a different place it needs to be \transcr{sup\schwa rscript}, but it isn't

And then it doesn't appear as it should in normal context anymore: \schwa.

\end{document}

You can also nest conditionals:

\documentclass[10pt]{article}
\usepackage{fontspec}
\usepackage{newunicodechar}

\newcommand{\transcr}[1]{%
  \textit{\transcrtrue #1}%
}
\newcommand{\phon}[1]{%
  \textbf{\phontrue #1}%
}

\newif\iftranscr % starts out false
\newif\ifphon % starts out false
\newunicodechar{ə}{%
  \iftranscr
    \textsuperscript{ə}%
  \else     
    \ifphon
      \textsubscript{/ə/}%
    \else
      ə%   
    \fi
  \fi
}
\newcommand{\schwa}{ə} % must be done _after_ \newunicodechar{ə}{...}

\begin{document}
In normal context, ə appears as it should.

In a different place it needs to be \transcr{supərscript}, and it is;        
likewise it could be \phon{supərscript}.

And then it doesn't appear as it should in normal context anymore: ə.

\end{document}

You need to exchange the order of the instructions \textit{#1} and \renewcommand{\schwa}{\textsuperscript{ə}}. In addition, both instructions need to be enclosed in a TeX group to limit the scope of the redefinition of \schwa.

\newcommand{\transcr}[1]{%
    {\renewcommand{\schwa}{\textsuperscript{ə}}%
    \textit{#1}}}

---

A full MWE:

\documentclass{article}
\usepackage{fontspec}

\newcommand{\transcr}[1]{%
    {\renewcommand{\schwa}{\textsuperscript{ə}}%
    \textit{#1}}}
\newcommand{\schwa}{ə}

\begin{document}
In normal context, \schwa\ appears as it should.

In a different place it needs to be \transcr{sup\schwa rscript}, and now it does.

And when it reappears in normal context --- \schwa\ --- things are still OK.
\end{document}