I managed to get it to work using the ExerciseList environment:
\documentclass[12pt]{article}
\usepackage[a4paper]{geometry}
\usepackage{mhchem}
\usepackage[lastexercise]{exercise}
\begin{document}
\section{Naming of Chemical Compounds - A ``DIY'' Tutorial}
\begin{ExerciseList}
\Exercise[title=DIY]
\ExePart{Question 1}
\Question{What is the symbol for copper?}
\Answer{\ce{Cu}}
\end{ExerciseList}
\end{document}
Use of \ExePart or redefinition of \QuestionNB
That said, you shouldn't need the \ExePart just to print "Question 1". It might be better to redefine \QuestionNB instead, like this:
\renewcommand{\QuestionNB}{Question~\arabic{Question}.\ }
Theorically, that should do what you want to achieve, although in practice, it's kind of a mess. I feel the reason is that the label is hardcoded in the package, and without a variable. This can however be "fixed" by resetting \QuestionIndent:
\setlength{\QuestionIndent}{7em}
So your example becomes:
\documentclass[12pt]{article}
\usepackage[a4paper]{geometry}
\usepackage{mhchem}
\usepackage[lastexercise]{exercise}
\begin{document}
\section{Naming of Chemical Compounds - A ``DIY'' Tutorial}
\renewcommand{\QuestionNB}{Question~\arabic{Question}.\ }
\setlength{\QuestionIndent}{7em}
\begin{ExerciseList}
\Exercise[title=DIY]
\Question{What is the symbol for copper?}
\Answer{\ce{Cu}}
\end{ExerciseList}
\end{document}
which works for me (although I would personally find it nicer to put questions on the next line -- note: I'm not using the mhchem package):

After playing a bit with more Questions/Answers, I do think it's missing a linebreak somewhere:

I'm also not very sure of the spacing: answers end up being closer to the next question than to the question they relate to...
Exercise title
In fact, it seems the problem with the linebreak comes from the fact that you're using \Exercise[title=DIY] instead of simply \Exercise{DIY}, which defines the exercise name. When I change this:
\documentclass[12pt]{article}
\usepackage[a4paper]{geometry}
\usepackage{mhchem}
\usepackage[lastexercise]{exercise}
\begin{document}
\section{Naming of Chemical Compounds - A ``DIY'' Tutorial}
\renewcommand{\QuestionNB}{Question~\arabic{Question}.\ }
\setlength{\QuestionIndent}{7em}
\begin{ExerciseList}
\Exercise{DIY}
\Question{What is the symbol for copper?}
\Answer{\ce{Cu}}
\Question{What is the symbol for aluminum?}
\Answer{\ce{Al}}
\Question{What is the symbol for iron?}
\Answer{\ce{Fe}}
\end{ExerciseList}
\end{document}
I get a linebreak:

and it is not necessary anymore to redefine \QuestionIndent:

Exerciseenvironment should probably end after theAnswerenvironment... that would make more sense to me. Just a guess though... – raphink Jul 05 '11 at 08:22See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help.
...
l.13 ...at is the symbol for copper?\end{Question}
? ` Any ideas?
– ptrcao Jul 05 '11 at 08:28