How to have multiple lines of intertext within align* environment?

You can use two possibilities:

• \intertext{text... \endgraf text...} (recommended)
• \intertext{text... \smallskip\newline\hspace*{\parindent} text...}

The accepted answer already shows the first possibility. Hence, I'll just show how the second possiblity looks like:

    \documentclass{article}
    \usepackage{amsmath,mathtools,amssymb}
    \DeclarePairedDelimiter{\sbar}{\lvert}{\rvert}
    \newcommand{\tsbar}[1]{\sbar{#1}^\sigma_s}
    
\begin{document}
    For this first one, I can't figure out an appropriate width for the parbox and I can't figure out how to indent the second line of text (neither the indent command nor an hspace works).
        \begin{align*}
    \tsbar{\phi x_1,\dots,v_k} 
        &= T \text{ iff } \langle \sigma(v_1),\dots,\sigma(v_k) \rangle \in \sbar{\phi}_s;\\
    \tsbar{\lnot \phi} 
        &= T \text{ iff } \tsbar{\phi} \neq T;\\
    \tsbar{\phi \rightarrow \psi} 
        &= T \text{ iff } \tsbar{\phi}\neq T \text{ or } \tsbar{\psi}=T;\\
    \tsbar{\forall x \phi} 
        &= T \text{ iff, for every } a\in D_s,\ \sbar{\phi}^{[v/a]} = T;\\
    \tsbar{\Box_> \phi} 
        &= T \text{ iff, for every } t > s, \tsbar{\phi} = T;\\
    \tsbar{\Box_< \phi} 
        &= T \text{ iff, for every } t < s, \tsbar{\phi} = T.\\
    \intertext{(Later-than is given the obvious definition: $t > s =_{df} s < t$.)
        \smallskip\newline
        \hspace*{\parindent}
               Then, the defined operators:}
    \tsbar{\Box_\geq \phi}  
        & = T \text{ iff, for every } t \geq s, \tsbar{\phi} = T;\\
    \tsbar{\Box_\leq \phi}  
        & = T \text{ iff, for every } t \leq s, \tsbar{\phi} = T;\\
    \tsbar{\Box \phi}       
        & = T \text{ iff, for every } t \leq s \vee s \leq t, \tsbar{\phi} = T.
        \end{align*}
    \end{document}

I'd use \linewidth for the width of the \parbox. There is a small catch: the macro called by align (or the *-variant thereof) doesn't accept \par inside it, so you can't use \par (or a blank line) for ending paragraphs in the \longintertext macro I suggest. A workaround is to use \endgraf.

\documentclass{article}
\usepackage{amsmath,mathtools,amssymb}
\DeclarePairedDelimiter{\sbar}{\lvert}{\rvert}

\newcommand{\tsbar}[1]{\sbar{#1}^\sigma_s}

\newlength{\normalparindent}
\AtBeginDocument{\setlength{\normalparindent}{\parindent}}
\newcommand{\longintertext}[1]{%
  \intertext{%
    \parbox{\linewidth}{%
      \setlength{\parindent}{\normalparindent}
      \noindent#1%
    }%
  }%
}

\begin{document}

For this first one, I can't figure out an appropriate width for the parbox and I can't figure out how 
to indent the second line of text (neither the indent command nor an hspace works).
\begin{align*}
        \tsbar{\phi x_1,\dots,v_k} &= T \text{ iff } \langle \sigma(v_1),\dots,\sigma(v_k) \rangle \in \sbar{\phi}_s;\\
        \tsbar{\lnot \phi} &= T \text{ iff } \tsbar{\phi} \neq T;\\
        \tsbar{\phi \rightarrow \psi} &= T \text{ iff } \tsbar{\phi}\neq T \text{ or } \tsbar{\psi}=T;\\
        \tsbar{\forall x \phi} &= T \text{ iff, for every } a\in D_s,\ \sbar{\phi}^{[v/a]} = T;\\
        \tsbar{\Box_> \phi} &= T \text{ iff, for every } t > s, \tsbar{\phi} = T;\\
        \tsbar{\Box_< \phi} &= T \text{ iff, for every } t < s, \tsbar{\phi} = T.\\
\longintertext{%
  (Later-than is given the obvious definition: $t > s =_{df} s < t$.)\endgraf
  Then, the defined operators:
}
        \tsbar{\Box_\geq \phi} &= T \text{ iff, for every } t \geq s, \tsbar{\phi} = T;\\
        \tsbar{\Box_\leq \phi} &= T \text{ iff, for every } t \leq s, \tsbar{\phi} = T;\\
        \tsbar{\Box \phi} &= T \text{ iff, for every } t \leq s \vee s \leq t, \tsbar{\phi} = T.
\end{align*}
\end{document}

\\ is defined for the alignment, but you can use \newline

\intertext{(The later-than relation on stages is given the obvious definition: $t > s =_{df} s < t$.)\newline
%
Then, the defined operators:}

(But you get a terrible line break with the text width in the MWE, just before the forced break)