The Answer is still no even if all variables have mean zero. Take $f(x)=x|x|$. Let $X$ take values $-5,-1,1,5$, equally likely, and let $Y$ take values $-4,4$, equally likely. Then $Y$ has greater variance than $X$ but this is reversed when you apply $f$ to both variables.

The Answer is still no even if all variables have mean zero. Take $f(x)=x|x|$. Let $X$ take values $-5,-1,1,5$, equally likely, and let $Y$ take values $-4,4$, equally likely. Then $Y$ has greater variance than $X$ but this is reversed when you apply $f$ to both variables. All this remains true for continuous variables as well. Add an independent variable $U$, uniform in $[-\epsilon, \epsilon ]$, to both $X$ and $Y$.