Well, if you can imagine typesetting anything, it can almost certainly be done in TeX. However, finding an elegant solution to this problem took a little work. So I'm hoping that this is the ouput you wanted to see:
Here's the code that produced that list (it adds 5 lines to your code, marked by the arrows):
\documentclass{article}
\usepackage{tikz-qtree-compat}
\usepackage{tikz}
\usepackage{stmaryrd}
\usepackage{gb4e}
\usepackage{booktabs} % <----
\begin{document}
\begin{exe}
\ex{Ann dances.}
\end{exe}
\begin{exe}
\ex
\begin{tabular}[t]{l}\toprule[-12pt] % <----
\begin{tikzpicture}[baseline=(current bounding box.center)]
\Tree [.S_0 [.NP [Bill ] ] [.VP [.V said ] [.NP \node(a){that}; ] ] ]
\begin{scope}[shift={(2in,0in)}]
\Tree [.\node(b){S_1}; [.NP Ann ] [.VP dances ] ]
\end{scope}
\draw[->](\subtreeof{a}.0) .. controls +(west:0) and +(west:1) .. (b);
\end{tikzpicture}
\\ \end{tabular} % <----
\end{exe}
\begin{exe}
\ex
\begin{tabular}[t]{l}\toprule[-12pt] % <----
$\left\llbracket
\begin{array}{l}
\Tree [ .S [.NP [.N Ann ] ] [.VP [.V dances ] ] ]
\end{array}
\right\rrbracket$
\\ \end{tabular} % <----
\end{exe}
\end{document}
Explanation:
Because exe is a gb4e-redefined list, and \ex is a gb4e-redefined \item, the first part of the job involved dropping each of your objects (tikzpicture and math-set array) into a \begin{tabular}[t]{l} environment, the [t] instructing tabular to set its baseline at the top.
All well and good, except that you need to issue an \hline somewhere in the environment to prod tabular into doing its work. That turns out to be problematic for two reasons: first, you probably don't want to see an ugly line across the top of your objects, and second, the \ex item number is now vertically aligned with the middle of the \hline, effectively quite some way above the top of each object. Not smooth.
Enter the booktabs package. It provides several \hline replacements (including \toprule, which we use). However, most wonderfully, you can can pass it an argument specifying the thickness of the line. So, \toprule[0pt] effectively eliminates the visual line. Even more wonderfully, you can pass it a negative number which acts like a negative \vspace, except without the problem that \vspace's can't be added here.
Voila. All that's left to do is to fine-tune \toprule's negative thickness to your exact requirements. Oh, and maybe add a little indentation to the "Ann dances" \ex to align it with your other objects (\ex{\quad Ann dances.} should do it). : )
