I've heard this claim disputed, and referred to as a CIA lie. The amount of partisan writing on this topic makes it difficult for a layperson to understand the facts.
I am unaware of Guevara executing any homosexuals because of their homosexuality. His diaries reveal that he did execute people, but these were for political reasons (like working with Batista and against the rebels), not because of their homosexuality. That isn't to say that some of the people he executed for political reasons couldn't have been gay, but that's obviously not what people pushing this argument are implying. You are 100% right about the partisanship here, since while the Cuban government has tried to move beyond its own documented persecution of gay people it is loath to cooperate too much in giving information on who did what, when, and why, since a lot of the upper echelons of the current government would be implicated.
During the first several decades of the Cuban Revolution there was undoubtedly a great deal of homophobia. Author Guillermo Cabrera Infante has talked about having gay men crash on his couch and floor because they felt unsafe at home or in the streets.
The Cuban Revolution never sent homosexual Cubans to death camps, as the Nazis had done, but instead sent them to forced labor camps called the UMAP between 1965-1968. Homosexuality was seen as a sign of bourgeois corruption and 'manly' work was supposed to reeducate them and make them men again. Treatment at the camps could be brutal and is still a stain on Cuba's record.
After a great deal of domestic and international pushback, Fidel Castro closed the camps in 1968. This was, however, the same year he began the Gran Ofensiva Revolucionario, grand revolutionary offensive, which nationalized the remainder of the private sector. It was also the same year that works by authors like Heberto Padilla and Anton Aruffat got their work published by the State publishing houses with warnings for their problematic ideological content.
In 1971 the Congress on Education and Culture cemented homophobia as a matter of government policy by prohibiting homosexuals from being in positions of power, such as teaching, from which they could feasibly 'recruit' people. It also inaugurated the so-called Grey Quinquennial (Quinquenio Gris), a five year period of extremist ideological orthodoxy and heightened homophobia. Some gay men, like Alfredo Guevara (no relation to Che), were in privileged positions, having known Fidel for years or having other important protectors, and were thus shielded from the worst of it. Most were not so lucky. Especially since one consequence of the mass nationalizations of 1968 was that those gay bars that did exist were turned over to the State, destroying what havens gay Cuban men had. (I will note here that most scholars, including myself, are more familiar with persecution of gay men than persecution against Lesbians, which seems to have been a lower priority for Cuban homophobes)
The 1980s were a bit more relaxed, or at least it was less of a matter of public policy to persecute gay people, but homophobia really started to break down in the 1990s with Fresa y Chocolate. It was originally a short story written by (heteroseuxal) Cuban author Senel Paz and was later turned into a film. In it, a Cuban member of the Communist Youth meets a gay Cuban man who works in the Culture Ministry and they develop a friendship. Not only is the gay man shown as a man, but also as a patriot who often knew more about the history and culture of the island than the supposedly rapidly nationalist Cuban Communist Youth member. In the end, the gay man sticks up for a gay artist friend and is expelled from all jobs related to culture, deciding to leave the country rather than face continued persecution. Both the short story and film were watershed moments in the Cuban discussion of LGBT rights and got the ball rolling on a change in the popular perception of gay people. This has been backed, more recently, by SENESEX, a Cuban governmental organization dedicated to sex education and LGBT rights, run by Mariela Castro, daughter of President Raul Castro. Fidel Castro also accepted responsibility for the camps in this interview.
This is a long preamble on homophobia in Cuba, which predates the Revolution, so that you'll understand the context of me saying that Che had little to do with it. The UMAP, the darkest moment for gay people in Cuba, occurred between 1965-1968. Che left Cuba in 1965 to fight in anti-colonialist wars abroad and died in 1967. He was not in charge of camps and his time in Cuba in 1965 was spent training and preparing to leave, not in Camaguey where most of the camps were.
After the Revolution came to power many Cubans were sent to paredon (a firing wall) due to their behavior under the Batista dictatorship, but they were not sent to the firing wall for being gay.
As this is not exactly my field of expertise, I'll point to the following with words of caution on each one (warning - all are in Spanish except the documentary, which has subtitles):
Fidel admitting responsibility: http://www.jornada.unam.mx/2010/08/31/mundo/026e1mun
Openly anti-Castro documentary which includes valuable interviews with LGBT people and self-exiled Cuban intellectuals: https://www.youtube.com/watch?v=oATGXqa69TA
Senel Paz' short story: http://www.uvm.edu/~ivaleria/LoboBosqueHombreNuevo.pdf
Major pro-Cuban government intellectual, Ambrosio Fornet, talking about the 'mistakes' of the Quinquenio Gris: http://www.rebelion.org/noticia.php?id=45857
Hi, you may be interested in this earlier thread