Is it legal for a HOA to turn away a tenant based on a 5 year old felony charge that he was never convicted of?

HOA's were invented so groups of people could set their own rules about how they wanted to live.  People literally give over very precious constitutional rights to HOA when they join a sub-division controlled by an HOA.  HOA's do have two general limitations.  The first is that any rules or regulations can't limit use or ownership based on race, gender, or some other protected class.  For example, they can't bar people from renting to Hispanics.  Unfortunately, persons with a criminal history are not considered a "protected class," so you're not going to get any relief from this limitation-- unless they only apply the restrictive rule to minority applicants-- but not to non-minority applicants.

The second general limitation is the rules, by-laws, or articles that establish the HOA.  This is the one written "checks and balance" system for an HOA.  If their by-laws do not give them the specific authority to set rules regarding the criminal background history of tenants, then they can't impose the rules.  If any of these documents do grant the authority-- then yes, they can exclude you from the property even if you were never convicted of the offense.  Your future landlord has two options depending on the situations described above.  If he does not believe that the written rules or documents give the HOA the power to make the rules they have, then he can file a lawsuit to have it declared a void or unauthorized restriction.  If the HOA does have the rule making authority, then your future landlord needs to get on the board and get the rules changed.  There may also be provisions in the by-laws for property owners to over-rides decisions by the HOA.  These latter options will depend strictly on how the HOA is structured and the nature of the documents which grant their power.