I once read that only a devotee is allowed to chant the Hare Krishna mantra. Is this true?
I've never read anything like this. The shastras, scriptures, define a devotee as one who chants the holy name sincerely, even once. If we accept that definition, then the idea that "only a devotee is allowed to chant the Hare Krishna Mantra" becomes clearer. This isn't to say that only people who are devotees can chant, but if one chants then one is a devotee, so therefore only devotees chant ;-).
That being said, there are different qualities of chanting. No matter how one chants, there is some benefit because Krishna's name is very powerful. Any soul that comes in contact with His name becomes purified and proceeds forward in his/her spiritual journey.
On the other hand, a person who sincerely chants the holy name with the intention of serving the Lord and/or getting to know the Lord through His name will achieve a deep relationship with the Lord and the Lord will also reciprocate deeply in that friendship. A person who chants carelessly or without sincere intent will also benefit, but only minimally. Their relationship with the Lord will develop slowly.
Chaitanya Mahaprabhu specifically offers the chanting of the maha-mantra as the prime benediction for humanity in this difficult age of Kali, (quarrel and hypocrisy ) He says that in Kali-yuga, the Supreme Lord has kindly invested all of His potencies in His holy name, and there are no hard-and-fast rules for chanting such names. In this regard, Srila Prabhupada says:
"The mention of such rules refers to kala-desha-niyama, or regulations of time and place. Normally there are strict regulations governing the time, season, place, conditions, etc., under which one may execute a particular Vedic ceremony or chant a particular mantra. However, one should chant the holy name of Krishna everywhere and at all times, twenty-four hours a day. Thus there is no restriction in terms of time and place. This is the meaning of Chaitanya Mahaprabhu's statement."
So chanting the Lord's names is especially meant to allow us to get to know Him through His name and by taking advantage of that process one is considered a devotee in the eyes of the Lord.
I hope this is helpful