Does the dedication "To my beloved John" need a comma after "beloved"?
It depends on how you want the dedication to read. There are two main options:
Here, beloved is a noun. This says that John is equivalent to your beloved. A fine point to this is that this implies that you have only one "beloved". If you happen to have multiple beloveds (or multiple Johns -- thank you, Kit), you can use beloved as an adjective by saying:
This second option used beloved as a modifier of John. If you have multiple Johns, this can help differentiate between them. Maybe you have a "beloved John", "tall John", and "carpenter John".
Either option is equally correct; the choice depends on how you feel about John.
The answer would be clearer with a complete sentence, but it is an introductory clause, and therefore needs a comma after 'John', but not after 'beloved'. Here, 'beloved' is an adjective, not a noun synonomous with 'John'.