Non-Forcing and Independence

The Gödel-Rosser sentence $R$ for $\text{ZFC}$ is an arithmetic assertion, such that $\text{ZFC}$ is equiconsistent with $\text{ZFC}+R$ and with $\text{ZFC}+\neg R$. So the Rosser sentence does not increase consistency strength. Since arithmetic assertions are preserved by forcing, one cannot use forcing directly to prove the independence of $R$.

Another example would be $\neg\text{Con}(ZFC)$, since $\text{ZFC}$ is equiconsistent with $\text{ZFC}+\neg\text{Con}(\text{ZFC})$, and so this doesn't increase consistency strength. (the theory $\text{ZFC}+\text{Con}(\text{ZFC})$, in contrast, does have strictly higher consistency strength). So this is an arithmetic assertion that is independent of $\text{ZFC}$, assuming that $\text{ZFC}$ is consistent, but this is not possible to prove in any direct way by forcing, since forcing does not affect arithmetic truth.

If we consider $ZF$ instead of $ZFC$, then we can say more. There are examples of such results which are obtained by Krivine, using his method of realizability. For example in the paper Realizability algebras II: New models of ZF+DC the following is stated:

Using the proof-program (Curry-Howard) correspondence, we give a new method to obtain models of $ZF$ and relative consistency results in set theory. We show the relative consistency of $ZF + DC$ + there exists a sequence of subsets of $\mathbb{R}$ the cardinals of which are strictly decreasing + other similar properties of $\mathbb{R}$.

As it is stated in the introduction of the paper:

These results seem not to have been previously obtained by forcing.

see also 50 years after forcing, the Curry-Howard correspondence gives new models of ZF