Do open embeddings preserve the intersection pairing?

I believe I just realized the answer is yes. The crux is that the Poincaré dual of a chain can be chosen to be supported in any arbitrarily small neighborhood supporting the chain. Thus, the Poincaré dual of an element in the image of $f_{\ast}$ can be chosen to be supported in the open subset $U=f(M)\subseteq N$, but then the preimage of this support under $f$ remains compact. Thus, while $f$ not being proper means that we don't get an induced map and a diagram like in the question post, we can mimic the diagram in terms of elements.

Lemma 1: Let $M$ be an oriented manifold, $U\subseteq M$ an open subset and $K\subseteq U$ be compact. Then, $U$ inherits the structure of an oriented manifold from $M$. If $j\colon U\rightarrow M$ is the inclusion, the fundamental classes obey the relationship $j_{\ast}[U,U-K]=[M,M-K]$.

Proof: Let $n=\dim(M)$. The diagram $$\require{AMScd} \begin{CD} H_n(U,U-K) @>>> H_n(M,M-K)\\ @VVV @VVV\\ H_n(U,U-x) @>>> H_n(M,M-x) \end{CD},$$ where each map is induced by inclusion, is commutative for each $x\in K$. The bottom map is an isomorphism identifying the local orientations by definition of the induced orientation of $U$. The claim follows.$\square$

Lemma 2: Let $M$ be an oriented manifold, $U\subseteq M$ be open and $j\colon U\rightarrow M$ the inclusion. If $\beta\in H_k(U)$, $K\subseteq U$ is compact and $\psi\in H^{n-k}(U,U-K)$ represents a Poincaré dual of $\beta$, then $(j^{\ast})^{-1}\psi\in H^{n-k}(M,M-K)$ represents a Poincaré dual of $j_{\ast}\beta\in H_k(M)$.

Proof: Since $U$ is an open subspace of $M$, it inherits the structure of an oriented manifold from $M$, so Poincaré duality on $U$ makes sense. Excision implies that $j^{\ast}\colon H^{n-k}(M,M-K)\rightarrow H^{n-k}(U,U-K)$ is an isomorphism, so the statement makes sense. Using Lemma 1, we calculate $$j_{\ast}\beta=j_{\ast}(j^{\ast}(j^{\ast})^{-1}\psi\cap[U,U-K])=(j^{\ast})^{-1}\psi\cap j_{\ast}[U,U-K]=(j^{\ast})^{-1}\psi\cap[M,M-K].$$ The claim follows.$\square$

Theorem: Let $f\colon M\rightarrow N$ be an orientation-preserving, open embedding between two oriented manifolds. Then $f$ preserves the intersection pairing.

Proof: Let's write $U=f(M)\subseteq N$, $j\colon U\rightarrow N$ for the inclusion and $\overline{f}\colon M\rightarrow U$ for the homeomorphism obtained by corestricting $f$, such that $j\overline{f}=f$. Let $\alpha\in H_k(M),\beta\in H_l(M)$ be arbitrary. Choose compact $K\subseteq U$ (resp. $L\subseteq U$) and $\varphi\in H^{n-k}(U,U-K)$ (resp. $\psi\in H^{n-l}(U,U-L)$), such that $\varphi$ (resp. $\psi$) represents a Poincaré dual of $\overline{f}_{\ast}\alpha$ (resp. $\overline{f}_{\ast}\beta$) by Poincaré duality. Now, because $K\subseteq U$ and $\overline{f}$ is a homeomorphism, $f^{-1}(K)=\overline{f}^{-1}(K)\subseteq M$ is compact. The cohomology class $\overline{f}^{\ast}\varphi\in H^{n-k}(M,M-f^{-1}(K))$ satisfies $$\overline{f}_{\ast}(\overline{f}^{\ast}\varphi\cap[M,M-f^{-1}(K)])=\varphi\cap\overline{f}_{\ast}[M,M-f^{-1}(K)]=\varphi\cap[U,U-K]=\overline{f}_{\ast}\alpha.$$ Since $\overline{f}$ is a homeomorphism, $\overline{f}^{\ast}\varphi\cap[M,M-f^{-1}(K)]=\alpha$. Thus, $\overline{f}^{\ast}\varphi$ represents the Poincaré dual of $\alpha$. Analogously, $\overline{f}^{\ast}\psi$ represents the Poincaré dual of $\beta$. On the other hand, $(j^{\ast})^{-1}\varphi$ (resp. $(j^{\ast})^{-1}\psi$) represents the Poincaré dual of $j_{\ast}\overline{f}_{\ast}\alpha=f_{\ast}\alpha$ (resp. $j_{\ast}\overline{f}_{\ast}\beta=f_{\ast}\beta$) by Lemma 2. Thus, $$\langle f_{\ast}\alpha,f_{\ast}\beta\rangle_N=((j^{\ast})^{-1}\varphi\cup(j^{\ast})^{-1}\psi)\cap[N,N-K\cap L]=(j^{\ast})^{-1}(\varphi\cup\psi)\cap[N,N-K\cap L].$$ On the other hand, $$\langle\alpha,\beta\rangle_M=(\overline{f}^{\ast}\varphi\cup\overline{f}^{\ast}\psi)\cap[M,M-f^{-1}(K)\cap f^{-1}(L)]=\overline{f}^{\ast}(\varphi\cup\psi)\cap[M,M-f^{-1}(K\cap L)].$$ Putting everything together, \begin{align*} f_{\ast}\langle\alpha,\beta\rangle_M&=j_{\ast}\overline{f}_{\ast}(\overline{f}^{\ast}(\varphi\cup\psi)\cap[M,M-f^{-1}(K\cap L)])\\ &=j_{\ast}((\varphi\cup\psi)\cap\overline{f}_{\ast}[M,M-f^{-1}(K\cap L)])\\ &=j_{\ast}((\varphi\cup\psi)\cap[U,U-K\cap L])\\ &=j_{\ast}(j^{\ast}(j^{\ast})^{-1}(\varphi\cup\psi)\cap[U,U-K\cap L])\\ &=(j^{\ast})^{-1}(\varphi\cup\psi)\cap j_{\ast}[U,U-K\cap L]\\ &=(j^{\ast})^{-1}(\varphi\cup\psi)\cap[N,N-K\cap L]\\ &=\langle f_{\ast}\alpha,f_{\ast}\beta\rangle_N.\end{align*}