I haven't heard of this, though I am sure anything could happen. I have heard quite a few stories of men feeling a burning sensation from it, though even that seemed few and far between. Have you tried having sex during your ring out week to see if he is able to hold the erection at that time? If he is, then you know for sure the culprit. If he isn't, then he may have a bit of erectile dysfunction, which is rather common for men. Be supportive and trying having intercourse during your out-week (possibly before your period starts to avoid any messiness) to see how it goes and then go from there.
erection burning sensation messiness having intercourse having sex