Monologue: a (usually long) dramatic speech by a single actor.
A fear of performing memorized speeches within plays or other performance types.
A fear of forgetting a memorized speech during a performance.
A fear that encompasses avoiding trying out for plays or other performance venues with long speeches in them, that must be memorized.
I can't try out for the community theater's version of Steel Magnolias, because I have monologophobia.
If I get the part of Hamlet, Prince of Denmark, then I am sure that my monologophobia will ruin my performance.