I had a fairly useless comment posted earlier which I apologize for. It turns out much is known though.  Below everything is restricted to zero applied field.

For a finite square grid of size $m$ by $n$ with periodic boundary conditions, the expression for the partition function $Z$ (from which the entropy per site can be worked out) was given already by [Kaufmann](https://doi.org/10.1103/PhysRev.76.1232) (note that you can find freely accessible copies of this paper through google) very shortly after Onsager's solution:
$$
\begin{split}
Z &=\frac{1}{2}\big(2\sinh(2H)\big)^{mn\over 2}\left\{\prod_{r=1}^n 2\cosh\Big(\frac{m}{2}\gamma_{2r-1}\Big)+\prod_{r=1}^n2\sinh\Big(\frac{m}{2}\gamma_{2r-1}\Big)\right. \\
&\quad\quad\qquad\qquad\qquad\qquad\left.+\prod_{r=1}^n2\cosh\Big(\frac{m}{2}\gamma_{2r}\Big)+\prod_{r=1}^n2\sinh\Big(\frac{m}{2}\gamma_{2r}\Big)\right\}
\end{split}
$$
where
* $H=J/k_BT$, $H'=J'/k_BT$, with $J,J'$ being the coupling constants in the horizontal and vertical directions, respectively, 
* $\cosh\gamma_j=\cosh(2H^*)\cosh(2H')-\sinh(2H^*)\sinh(2H')\cos(\pi j/n)$ and 
* $H^*$ is defined to as the solution to the equation 
$\tanh H^*:= \exp(-2H)$.

A fuller explanation is in chapter IV of McCoy and Wu's book "The Two-dimensional Ising model".  Another way to derive the partition function which has proved useful for [generalizations to antiperiodic boundary conditions](https://doi.org/10.1088/0305-4470/35/25/304) was worked out in [this paper by V.N. Plechko](https://doi.org/10.1007/BF01017042).

On an infinitely long cylinder with circumference $L$ I was able to find in a [2004 paper by Huang et al](https://arxiv.org/abs/cond-mat/0407731v3) an expression for the free energy (with $J=J'=1$):
$$
f=-\frac{1}{2}\ln(4z)-\frac{1}{2L}\sum_p\int_0^{2\pi}\frac{d\phi}{2\pi}\ln\left[z+z^{-1}-\Phi_p(\phi)\right]
$$
where now 
* $z=\sinh(2\beta)$ and 
* $\Phi_p(\phi)=\cos\phi+\cos\frac{2\pi p}{L}$.