It isn't entirely clear which line you are trying to draw: a line from the left to right of the page? a diagonal? (which one?)
This example uses colours to distinguish four possibilities:

\documentclass{article}
\usepackage{tikz}
\begin{document}
\thispagestyle{empty}
\begin{tikzpicture}[remember picture,overlay]
\coordinate (A) at (current page.north west);
\coordinate (B) at (current page.south east);
\draw[line width=1.5pt, cap=round, red]
(A) -- (B);
\draw[line width=1.5pt, cap=round, blue]
([yshift=-15pt]current page.north west) -- ([yshift=-15pt]current page.north east);
\draw[line width=1.5pt, cap=round, green]
([yshift=15pt]current page.south west) -- ([yshift=15pt]current page.south east);
\draw[line width=1.5pt, cap=round, magenta]
(current page.south west) -- (current page.north east);
\end{tikzpicture}
\end{document}
The errors were likely caused by the two problems identified in comments:
- The final
\draw command must end in a terminating semicolon.
(current page.north,current page.north) is not a location. It confuses the (<x>,<y>) syntax with the (<named coordinate>) syntax. If you need something like this, you can say (<named coordinate 1> -| <named coordinate 2>) which will use the x bit of <named coordinate 1> and the y bit of <named coordinate 2>. In the case of the current page coordinates there isn't much point because the relevant points are all defined anyway.
However, for purposes of demonstration:
\documentclass{article}
\usepackage{tikz}
\begin{document}
\thispagestyle{empty}
\begin{tikzpicture}[remember picture,overlay]
\draw[line width=1.5pt]
(current page.south west) .. controls (10,-10) .. (current page.north -| current page.east);
\end{tikzpicture}
\end{document}

(A) -- (B). – cfr Aug 28 '15 at 11:15(current page.north east, current page.north east)does not make sense, It is sort of like((a,b),(c,d)(current page.north east)is a coordinate on the page – daleif Aug 28 '15 at 11:17