Processing the original code:
\documentclass{article}
\usepackage{tikz}
\begin{document}
\begin{tikzpicture}
\draw[gray] ++(150:2.3) -- (0,0); %hypotenuse
\draw[teal] ++(180:2) -- (0,0); %adjacent
\draw[orange] (-2,1.15) -- (-2,0); %opposite
\draw[thin] (-0.5,0.25) arc (150:180:0.5)
node[left] {\small $30^\circ$};
\end{tikzpicture}
\end{document}
and zooming the resulting object, reveals two additional problems: the arc doesn't touch the hypotenuse and the hypotenuse and the shorter cathetus don't intersect:

Both problems are due basically to the same reason: manual calculation of coordinates.
To solve the problem with the arc, I would suggest you to use clipping and an additional node to draw the circle; in this way, you solve two problems: the arc is correctly drawn in an automated way (no need to guess angles as in the other answers) and you gain finer control over the label position:
\documentclass{article}
\usepackage{tikz}
\begin{document}
\begin{center}
\begin{tikzpicture}
\draw[gray] ++(150:2.3) -- (0,0); %hypotenuse
\draw[teal] ++(180:2) -- (0,0); %adjacent
\draw[orange] (-2,1.15) -- (-2,0); %opposite
\path[clip] (0,0) -- (-2,0) -- (-2,1.15) -- cycle;
\node[circle,draw,minimum size=40pt] at (0,0) (circ) {};
\node[font=\footnotesize,left] at (circ.160) {$30^\circ$};
\end{tikzpicture}
\end{center}
\end{document}

There's still one other detail to improve: the hypotenuse and the opposite cathetus not intersecting. This can be solved by using the intersections library to automatically do the calculation of the intersection between the prolongations of the opposite cathetus and the hypotenuse:
\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{intersections}
\begin{document}
\begin{center}
\begin{tikzpicture}
% place coordinates at the two initial vertices
\coordinate (a) at (0,0);
\coordinate (b) at (-2,0);
% automatically calculate the third vertex
\path[name path=line 1] (-2,0) -- (-2,2);
\path[name path=line 2] (0,0) -- (150:2.4);
\path [name intersections={of=line 1 and line 2, by={c}}];
% draw the lines
\draw[gray] (a) -- (c); %hypotenuse
\draw[teal] (a) -- (b); %adjacent
\draw[orange] (b) -- (c); %opposite
% draw the arc clipping a circle against the triangle and place the label
\path[clip] (a) -- (b) -- (c) -- cycle;
\node[circle,draw,minimum size=40pt] at (0,0) (circ) {};
\node[font=\footnotesize,left] at (circ.160) {$30^\circ$};
\end{tikzpicture}
\end{center}
\end{document}
