I suffered from ME from the ages of 14-18, and overdoing it (exactly like your boyfriend, I wouldn't take advice from anyone about cutting down) ended up making me 10x as bad and I had to be pulled out of school on medical grounds. I ended up missing all of my crucial exams after trying to force myself through my standard grades and then had to take an obscure route to finally get into uni when I had recovered. My advice would be to take it easy, revise little and often and avoid exerting yourself. Eating very clean foods also made a slight difference, so if he doesn't already have a healthy diet, then maybe suggest that. Unfortunately, there isn't really much he can do except ride it out. It's an awful, frustrating and painful experience, but I promise it gets better.
Don't let him believe that these exams are the be all and end all of his life. Let him know that there are other routes if university is his end goal, but he has to be healthy again before that can happen. Now that I'm finally at uni, I know I certainly wouldn't be able to cope if I was still suffering with ME, so although people look at me funny when I tell them I'm 20 and in first year, I'm entirely grateful that I was given the chance to recover before attending uni.
Sorry that was a bit long and babbly!