Runaway argument? with \newcommand[2]

\expandafter only expands one token after the token that follows (unless there are arguments). In your case this is just #1. One way to get your result, is to collect #1 #2 expanded in a macro before insertion:

\documentclass{article}

\makeatletter

% Timestamp
\newcommand{\timeStamp}[2]{%{#1-Date (YYYY.MM.DD)}{#2-Time (HH:MM)}
  \edef\mytmp{#1 #2}\expandafter\timeStamp@t\mytmp\@nil%
}%
\def\timeStamp@t#1.#2.#3 #4:#5\@nil{%
    \the\numexpr#5+#4*60+(#3-1)*60*24+(#2-1)*60*24*31+(#1-2017)*60*24*31*365\relax%
}%

\makeatother

\begin{document}
\tracingmacros=2\tracingcommands=2
\def\tOne{03:00}%
\def\dOne{2017.08.01}%
\timeStamp{\dOne}{\tOne}
\end{document}

This approach should generalise well to more than two arguments.

One can do some juggling with \expandafter, but a direct approach may be better:

\documentclass{article}
\usepackage{xparse}

\ExplSyntaxOn
\cs_new:Nn \joseph_time_stamp:nn
 {
  \__joseph_time_stamp:w #1.#2\q_stop
 }
\cs_generate_variant:Nn \joseph_time_stamp:nn { ff }
% a devious trick for the colon
\use:x
 {
  \cs_new:Npn
   \exp_not:N \__joseph_time_stamp:w
   ##1.##2.##3.##4\token_to_str:N :##5
   \exp_not:N \q_stop
 }
 {
  \int_eval:n
   {
    #5+#4*60+(#3-1)*60*24+(#2-1)*60*24*31+(#1-2017)*60*24*31*365
   }
 }
\NewExpandableDocumentCommand{\timeStamp}{mm}
 {
  \joseph_time_stamp:ff { #1 } { #2 }
 }
\ExplSyntaxOff

\begin{document}

\def\tOne{03:00}
\def\dOne{2017.08.01}

\timeStamp{\dOne}{\tOne}

\timeStamp{2017.1.1}{0:0} % should print 0

\end{document}

If the time specification didn't include the colon, it would be easier; the problem is that : is special in the scope of \ExplSyntaxOn, so the direct

\cs_new:Npn \__joseph_time_stamp:w #1.#2.#3.#4:#5\q_stop
 {
  \int_eval:n
   {
    #5+#4*60+(#3-1)*60*24+(#2-1)*60*24*31+(#1-2017)*60*24*31*365
   }
 }

would not work and an indirect method is needed in order to “stringify” the colon.

An expandable solution with \expandafter and argument juggling (OS for “old style”):

\makeatletter
\newcommand{\timeStampOS}[2]{%
  \expandafter\timeStampOS@a\expandafter{#2}{#1}%
}
\newcommand{\timeStampOS@a}[2]{%
  \expandafter\timeStampOS@b\expandafter{#2}{#1}%
}
\newcommand{\timeStampOS@b}[2]{\timeStampOS@c #1 #2\@nil}
\def\timeStampOS@c #1.#2.#3 #4:#5\@nil{%
  \the\numexpr
    #5+#4*60+(#3-1)*60*24+(#2-1)*60*24*31+(#1-2017)*60*24*31*365
  \relax
}
\makeatother

expand #1 and #2

% Timestamp
\newcommand\timeStamp[2]{%
    \expandafter\timeStamp@t#2 #1\@nil}%
\def\timeStamp@t#1:#2 #3\@nil{\expandafter\timeStamp@@t#3 #1:#2\@nil}%
\def\timeStamp@@t#1.#2.#3 #4:#5\@nil{%
  \the\numexpr#5+#4*60+(#3-1)*60*24+(#2-1)*60*24*31+(#1-2017)*60*24*31*365\relax%
}%