Easy case with uncolored PDF and pdfTeX (unhappily very seldom):
The PDF file contains some drawings without explicit color settings and
the image will be included by the pdftex driver of the graphics package.
The following file generates such an image:
\documentclass{article}
\usepackage[active,tightpage,floats]{preview}
\begin{document}
\begin{figure}
\Huge\bfseries\sffamily
\setlength{\fboxrule}{2pt}
\fbox{Hello World}
\end{figure}
\end{document}

The following file includes the image (t.pdf) and changes its colors.
\documentclass{article}
\usepackage{color}
\usepackage{graphicx}
\pagecolor{black}
\color{white}
\begin{document}
\Huge\bfseries
\noindent
Some text.\\
\includegraphics{t.pdf}\\
Some text.\\
\textcolor{yellow}{\includegraphics{t.pdf}}\\
Some text.
\end{document}

This trick does not work with other drivers, because the other drivers
normalize the color before the image is included. If pdftex.def is given
the option resetcolor
then the color is set to \normalcolor
during
image inclusion.
Manual color fixes
With some knowledge of the internals structures the PDF file can be fixed
manually. First the compressed page contents needs to be uncompresed, e.g.:
pdftk test.pdf cat output test-uncompress.pdf uncompress
Then the pages content streams need to be identified and the color operations
can be either changed or deleted (by overwriting with spaces). Best is not to
change the size of the object otherwise the file offsets of the objects in
the xref table needs to be updated. BTW, also PDF (or PS) use %
as comment char.
Then the file can be recompressed.
Or the PDF file is converted to PostScript (e.g. with pstopdf from xpdf). Editing
is easier if changing the file size does not matter.
PostScript does not have a xref table. However, the detection of
the color operators can be more difficult, because often they are renamed or
hidden in procedures. Unlike PDF, PostScript is a programming language.
Programs for processing vector images like Inkscape
If the PDF file can be successfully imported in Inkscape or similar programs,
the colors could be changed there.
Bitmap conversion
As last resort the image can be converted to a bitmap image (ghostscript and other convertes). That means quality
loss because of pixel data. But many image processing programs should be able
to change the colors.
Generating inverse colors can even be done by a feature of the PDF format that
is supported by pdftex.def. Colors can be inverted via the /Decode
array (except
images with indexed color spaces). Each
color component has then two float values inbetween 0 and 1.
Thus the number of color components must be known (Mono: 1, RGB: 3, CMYK: 4)
\includegraphics[decodearray=1 0 1 0 1 0]{rgbimage.png}
For example, the ghostscript device png16m
can be used to generate PNGs for
this usage. More details are explained in the PDF reference. The options of
pdftex.def are shortly explained in the file itself.