3 Typo corrected

UPDATE: I have updated this answer slightly to take into account Victor's remark.

I think that the precise questions being asked admit a straightforward answer. At the moment, no such formula is known and the proofs of Gross-Zagier, Waldspurger, Zhang et al. and Howard all absolutely and crucially require the hypothesis of self-duality. The reason for this is that the representation-theoretic part of the proof requires an understanding of test-vectors, as in the works of Tunell and Saito or as in the conjecture of Gross-Prasad. This is explained for instance in the article of Gross entitled Heegner points and representation theory as well as in Non-triviality of CM points in ring class field towers. Aflalo, Esther and Nekovář, Jan. Israel J. Math. 175 (2010), 225--284 (in which the formal setting is explored).

As for whether similar formula could hold, I am not too optimistic. A Gross-Zagier formula should involve the $\psi$-eigenpart of the action of $\textrm{Gal}(H/\mathbb Q)$ on the projection of a CM point on the $\pi(f)$-component of the Jacobian of a Shimura curve. However, comparing the Galois action on CM points with the adelic action on the Jacobian, we see that this $\chi$-eigenpart can be non-trivial only when the restriction of $\psi$ to $\mathbb A_{\mathbb Q}$ is equal to $\chi$, or equivalently only in the self-dual case. This is proved for instance in Cornut, Christophe; Vatsal, Vinayak Nontriviality of Rankin-Selberg L-functions and CM points. Lemma 4.6.

Note also that this is what we should expect from the conjectures on special values of $L$-functions applied to $L(f/K,\psi,s)$ when $f$ is not self-dual. In that case, the conjecture implies that $L'(f/K,\psi,s)$ should be related to cohomology classes in the dual of the motive of $f$. Only in the self-dual case does these collapse in a formula involving the height of a point. Finally, in the situation you describe, even though $L(f,\psi,s)$ might vanish at 1, it is is expected that it doesn't generically (say in a relevant $\mathbb Z_{p}$-extension), so the conjectural relation between $L(f/K,\psi,s^$ L(f/K,\psi,s)$(or its Selmer group) and putative point could hold only "locally at the specialization corresponding to$f$" in a$p$-adic family of automorphic representation containing$\pi(f)\otimes\psi$. All the arguments that I know relating these objects would then simply vanish. Now an argument from ignorance is not a very good one, and I would very much like to be proven false, if only to learn something. Hidden behind all this is the question of the link between Kato's Euler system and rational points on modular varieties. The link is mysterious to me, but David Loeffler and Sarah Zerbes have some ideas. 2 Took into account comments and expanded the answer UPDATE: I have updated this answer slightly to take into account Victor's remark. As for whether similar formula could hold, I am not too optimistic. A Gross-Zagier formula should involve the$\psi$-eigenpart of the action of$\textrm{Gal}(H/\mathbb Q)$on the projection of a CM point on the$\pi(f)$-component of the Jacobian of a Shimura curve. However, in comparing the sense that if it doesGalois action on CM points with the adelic action on the Jacobian, it will be for completely different reasons than those I we see that this$\chi$-eigenpart can understand. Looking at be non-trivial only when the problem from an angle I actually kind restriction of know, if such formula were$\psi$to hold$\mathbb A_{\mathbb Q}$is equal to$\chi$, or equivalently only in the self-dual case. This is proved for instance in Cornut, then combining them with conjecture Christophe; Vatsal, Vinayak Nontriviality of Rankin-Selberg L-functions and CM points. Lemma 4.6. Note also that this is what we should expect from the conjectures on special values of$L$-functions, you L$-functions applied to $L(f/K,\psi,s)$ when $f$ is not self-dual. In that case, the conjecture implies that $L'(f/K,\psi,s)$ should be able related to relate Selmer groups and Heegner points cohomology classes in non the dual of the motive of $f$. Only in the self-dual situations, but that seems rather hopeless using current techniquescase does these collapse in a formula involving the height of a point. To start withFinally, in the situation you describe, even though $L(f,\psi,s)$ might vanish at 1, it is is expected that it doesn't generically (say in a relevant $\mathbb Z_{p}$-extension), so the conjectural relation between your $L(f/K,\psi,s^$ (or its Selmer group) and putative Heegner points point could hold only "locally at the specialization corresponding to $f$" in a $p$-adic family of automorphic representation containing $\pi(f)\otimes\psi$. This means that all All the arguments that I know to relate Selmer groups to Heegner points relating these objects would then simply vanish.

Now an argument from ignorance is not a very good one, and I would very much like to be proven false, if only to learn something. Hidden behind all this is the question of the link between Kato's Euler system and rational points on modular varieties. The link is mysterious to me, but David Loeffler and Sarah Zerbes have some ideas.

1

I think that the precise questions being asked admit a straightforward answer. At the moment, no such formula is known and the proofs of Gross-Zagier, Waldspurger, Zhang et al. and Howard all absolutely and crucially require the hypothesis of self-duality. The reason for this is that the representation-theoretic part of the proof requires an understanding of test-vectors, as in the works of Tunell and Saito or as in the conjecture of Gross-Prasad. This is explained for instance in the article of Gross entitled Heegner points and representation theory as well as in Non-triviality of CM points in ring class field towers. Aflalo, Esther and Nekovář, Jan. Israel J. Math. 175 (2010), 225--284 (in which the formal setting is explored).

As for whether similar formula could hold, I am not too optimistic, in the sense that if it does, it will be for completely different reasons than those I can understand. Looking at the problem from an angle I actually kind of know, if such formula were to hold, then combining them with conjecture on special values of $L$-functions, you should be able to relate Selmer groups and Heegner points in non self-dual situations, but that seems rather hopeless using current techniques. To start with, in the situation you describe, even though $L(f,\psi,s)$ might vanish at 1, it is is expected that it doesn't generically (say in a relevant $\mathbb Z_{p}$-extension), so the relation between your Selmer group and putative Heegner points could hold only "locally at the specialization corresponding to $f$" in a $p$-adic family of automorphic representation containing $\pi(f)\otimes\psi$. This means that all the arguments that I know to relate Selmer groups to Heegner points would simply vanish.

Now an argument from ignorance is not a very good one, and I would very much like to be proven false, if only to learn something.