Your problem is caused by the second byte of the problematic characters.
Let me first name a few characters that replicate the problem (hex code in the parenthesis is the code point of the corresponding character in Big5 encoding).
! File ended while scanning use of \CJK@XX.: 認 (0xBB7B), 程 (0xB57B), and 臨 (0xC17B)
! Argument of \CJK@XX has an extra }.: 開 (0xB67D), 稀 (0xB57D), and 飼 (0xB97D)
You might notice that all the characters that cause ! File ended while scanning use of \CJK@XX. have 0x7B in their second byte. In ASCII code, 0x7B means {, which has a special meaning to LaTeX. So, a part of 認 was interpreted as an unbalanced opening brace, hence the error. Same explanation can be applied to ! Argument of \CJK@XX has an extra }. because 0x7D in ASCII code means }.
Sometimes, more weird thing can happen. The following code can be typeset without an error, but the output is totally unexpected (Big5-encoded of course).
\documentclass{article}
\usepackage{CJK}
\begin{document}
\begin{CJK*}{Bg5}{bsmi}
臨開
\end{CJK*}
\end{document}

Considering the fact that 褻 has the code point 0xC1B6 in Big5 encoding, it's no wonder that this happens.
In order to overcome this restriction, CJK package prepares a utility program bg5conv along with sjisconv (for Shift_JIS) and extconv (for Big5+ and GBK). These programs convert characters into a safe form so that they are compatible with LaTeX.
Typeset the following code with -shell-escape option, and you will get a desired output.
\RequirePackage{filecontents}
\begin{filecontents}{\jobname-input.tex}
認程臨開稀飼
\end{filecontents}
\immediate\write18{bg5conv <\jobname-input.tex> \jobname-processed.tex}
\documentclass{article}
\usepackage{CJK}
\begin{document}
\begin{CJK}{Bg5}{bsmi}
\input{\jobname-processed.tex}
\end{CJK}
\end{document}

When you have the luxury, you can also use a shell script wrapper called bg5pdflatex, which converts the whole source and then runs pdfLaTeX. This doesn't require the -shell-escape option as is in Ch'en Meng's comment.
% Typeset this with bg5pdflatex or bg5latex.
\documentclass{article}
\usepackage{CJK}
\begin{document}
\begin{CJK}{Bg5}{bsmi}
認程臨開稀飼
\end{CJK}
\end{document}
However, I'm not sure if these are satisfactory solutions to the OP and I find them inconvenient.
\usepackage[T1]{fontenc}and compile. What output do you get then? – Ulrike Fischer Dec 29 '17 at 17:36