Quick solution:
Use \textcopyright instead of \copyright. Now the copyright sign will be ouput in cmsy font, and there will be a font warning.
This approach is somewhat bad because it produces a font substitution warning, and the use of CM fonts.
The advantage is, you don't need to modify the code much. And the font is exacly Palatino with its mathematical support provided by PSNFSS bundle.
Quick solution 2:
\renewcommand*\copyright{{%
\usefont{EU1}{lmr}{m}{n}\textcopyright}}
It use the sign in Latin Modern fonts.
This prevents the warnings. And use LMR font for this sign. The font command can be changed as you wish. e.g. \fontfamily{lmr}\selectfont.
textcomp package uses an old set of fonts, in TS1 font encoding.
Suggested solution:
\documentclass{minimal}
\usepackage{unicode-math}
\setmainfont[Ligatures=TeX,Numbers=OldStyle]{TeX Gyre Pagella}
\setmathfont{Asana Math}
\begin{document}
Test \copyright 2011
\end{document}
Now all fonts are changed to OpenType ones. The result is somewhat similar with pdfLaTeX + pxfonts package.