Boy's crime

Since it was implied throughout the movie that Iris's brother Gerald and her husband were having an affair, I would think the "crime" had something to do with homosexuality.

I was going to say his crime was designing that hideous dress that covers Constances beautiful face. Apparently in the original play he is a thief and she has syphilis. I do like the gay angle others posted.

When there are two, one betrays-Jean-Pierre Melville

It was, without a doubt, homosexuality. In Britain, especially in 1934 (actually anytime before the late 1960's in most of the English-speaking world), homosexuality was considered "unspeakable" and was a criminal offense. Although I agree with most of the first reviewer's summation of the plot, his conclusion that Boy's secret was venereal disease is utter rubbish. Having VD may well have carried a social stigma, but the note given to Iris clearly stated that Boy had been convicted under an assumed name for a "disgusting and repugnant" offense, which was how same-sexual contact was often described and treated by the British-based justice systems in pre-1970's. The fact that Iris was willing to "never speak of it again" is further proof that disease was not the secret as VD would have unlikely been incurable in those days and they would most definitely have to speak of it during the course of their marriage.

The other telling clue was that the camera deliberately pans on two separate occasions (near the end of the film) to the other set of initials carved together on the other tree trunk --that of Boy and Gerald. This was the director's blunt method of stating the secret, in the chance that the overly-subtle plot indicators hadn't conveyed the message earlier to the audience.