Vertical placement of math symbol in LaTeX

Try \mathchoice (or \mathpalette):

\newcommand{\myop}{%
  \mathchoice{\raisebox{1pt}{$\displaystyle\otimes$}}
             {\raisebox{1pt}{$\otimes$}}
             {\raisebox{0.5pt}{$\scriptstyle\otimes$}}
             {\raisebox{0.2pt}{$\scriptscriptstyle\otimes$}}}

The amsopn documentation says:

... the vertical spacing may not be optimal in script and scriptscript sizes. Unfortunately TeX provides no easy way to do math mode vertical spacing that varies with current math style like mu units.

$F_{\raisebox{1pt}{\scriptsize$\otimes$} I}$