I'm not too sure about persecution as Asexuality is a modern construction that from what I know of has only been recognised recently with the first group to advocate Asexuals and their rights, Asexual Visibility and Education Network ("AVEN") being setup recently. (the earliest mention of Asexuality that I know of its in 1896)

So in 1896 Sexologist and advocate for the rights of sexual minorities, Magnus Hirschfeld mentions in his pamphlet Sappho und Sokrates ("Sappho and Socrates") about Asexuality, he has this to say about it:

There are individuals who are without any sexual desire (“Anästhesia sexualis”)

Anästhesia sexualis ("Sexual Anesthesia") is a psychological term describing the lack of "normal" sensations when engaging in sex. However it can be used to describe the lack of wanting to have sex in general. This usually has a psychogenic background to it ("relating to anything to do with the mind").

Hirschfeld goes on to say this about the libido and how one would go about stimulating it but if one doesn't have one then one shouldn't be made to pretend:

It is also not possible to artificially evoke the kind of drive, that is not existent or almost not noticeable. In case of a complete atrophy there is no way that it would spontaneously develop. However, if there already is a mental disposition towards it, it can be increased by the upbringing, exercise, example, seduction and suggestion. We can wake up a sleeping drives, as long as their sprouts are big enough, but we can never create new ones.

Hirschfeld also uses the term, Automonosexualism which he uses whilst explaining Asexuality (even though you can experience both without experiencing the other). Automonosexualism is where can experience sexual satisfaction solely through masturbation only and in some cases regarding men, Autofellatio.

Some people (including Hirschfeld) have claimed that Automonosexualism is rare and transsexuals are most likely to experience this (Hirschfeld;1948. Blanchard; 1989, 1991. Freund and Blanchard; 1993) another phenomena supporting this claim is the term, Autogynephilia, where a man is sexually attracted to himself but he views himself as a woman.

The idea of Asexuality and Automonosexualism can be taken one step further when looking at Blanchard's Transsexualism Typology (which he devised in the 1980's & 1990's using Kurt Freund's work as a basis). In this theory, Blanchard divides Transwomen (MtF) into two categories; 1) Those who transition to attract men ("homosexual transsexuals") & 2) Those who transition because they're aroused by the thought of being Female ("autogynephilic transsexuals") they might only be aroused by that thought and not the prospect of having sex as a woman.

However gender identities and sexual orientations are never 100% clear cut its different for everyone. As far as Persecution goes, off the top of my head, I can't remember hearing of any that were persecuted as I say its a modern construction even in 2015 there is still dispute over the definition of Asexuality.

Hopefully this helps, if you have any follow-ups please don't hesitate to ask :)

Sources and Further Reading:

1. Sappho und Sokrates ~ Magnus Hirschfeld

2. Transsexualism Typology ~ Ray Blanchard

3. Understanding Asexuality ~ Anthony F. Bogaert