$\def\smat#1{\left(\begin{smallmatrix}#1\end{smallmatrix}\right)}$

The answer to your question is "no". Let $G=GL(2,\mathbb C) \times GL(2,\mathbb C)$ act on $X=GL(2, \mathbb C)$ by $(A,B)\cdot C= ACB^{-1}.$ The orbit of the matrix $x=\smat{
    1 & 1 \\
    0 & 1 \\\
}$
 is $X$. (Hence it is closed.) Let $H$ be the subgroup of matrices of the form
$\smat{
    a & 0 \\
    0 & a^{-1}\\\
}
\times
\smat{
    a & 0 \\\
    0 & a^{-1}\\\
}
\subset G$. Then $Hx=
\smat{
    1 & b \\\
    0 & 1\\\
}$, for $b\ne 0$. Hence $Hx$ is not closed.