[*Spoilers for various stories including: Arkham Asylum: A Serious House On Serious Earth (1989), World War Hulk (2008), Batman: Face The Face (2006).]
Superheroes, like real people, need therapy. However, I am putting forth that therapy can never fully be performed in superhero comics. This is mainly because a lot of heroes and villains in the superhero genre derive their motivations from tragedies. Two examples are Peter Parker a.k.a. Spider-Man and Bruce Wayne a.k.a Batman (Amazing Fantasy #15  and Detective Comics #33 ). Both of these characters’ motivations are the deaths of their parental figures. Thus we can presume that if treated with talk therapy, Peter Parker may not have developed psychological problems, such as his survivor’s guilt. While Bruce Wayne may not have become Batman if he had the right therapist and committed to the therapy.
Yet those two examples only show how therapy could stop a superhero career from starting. What about after the superhero’s, or villain’s, career has started? For this we should look at Bob Reynolds a.k.a. The Sentry/The Void and Harvey Dent a.k.a Two-Face. In the case of Reynolds, he has stated he has therapy-related diagnoses, but these sessions are never really shown (World War Hulk). While in Dent’s case his therapy sessions are shown and even had some success, but he always goes back to being Two-Face (Arkham Asylum: A Serious House On Serious Earth and Batman: Face The Face). Both Reynolds and Dent’s premises are dependent on having a split personality, therefore they can never be fully cured.
Yet there are exceptions where therapy has been successfully applied to superheroes. A primary example of therapy being successfully combined with the superhero genre is Tony Stark/Iron Man. In Iron Man’s case, we need look no further than at his recovery from alcoholism as a successful form of therapy. Said therapy, to my knowledge, happens through Stark mainly going to Alcoholics Anonymous meetings to recover (Invincible Iron Man Omnibus Vol. 1 ). Though Stark has had relapses at times, this could still be considered successful therapy (Iron Man #172 ).
(Iron Man #172  Written by Denny O’Neil, Penciled by Luke McDonnell, Inked by Steve Mitchell, Colored by Bob Sharen, Lettered by Rick Parker.)
In conclusion, therapy can work in the short form in the superhero genre. However, successfully dealing with psychological issues in the long-term risks negating a character’s motivation, premise, or super career. Therefore I posit that only a finite story can truly explore psychological issues in this genre.