If I remember correctly, the original post also stated that this only
happened using apacite.sty.
I've pared the sequence to this:
1) In LyX, export a latex file: File -> Export -> LaTeX.
2) Open the whatever.tex file with your favorite text editor.
3) Remove the line: \usepackage{babel}.
It should work correctly, unless you are using acknowledgments style.
There is a typo in the apa.cls. on line 457:
\long\def\acknowledgements#1{\long\def\@acks{#1}}
Unfortunately, there is not supposed to be an "e" after the "g" in
acknowledgments (acknowledgements vs. acknowledgments). Removing the
offending "e" mostly solves the problem. You should also edit the
whatever.tex file such that \title, \author,\affiliation, \shorttitle,
\abstract, and \acknowledments (in any order, I think) all precede
\begin{document} and \maketitle (in that order as per apacls.txt).
Additionally, you can not have a blank line following \maketitle.
From there proceed with:
latex whatever
bibtex whatever
latex whatever
latex whatever
xdvi whatever.dvi
Note that acknowledgments are only used in the journal layout option
(Layout -> Document ... -> Extra Options -> jou.
