Interesting timing for this question...
On the site, over the last few days, there are several related posts. There is some ASMI info with experts calling for taking at least 3 months off some time during the year. There's an interesting piece on Trevor Bauer and how hard and frequently he works and throws. There's last Sunday's HSBBW radio blog with esteemed coach Fred Corral saying he thinks you need to keep throwing. There are several references to pro's and con's of long toss. And, as I type, I'm listening to LL regional playoffs where a kid from Alabama went down in a heap after throwing a pitch.
Many experts are stating with conviction opposite theories. So, good luck getting a solid and consistent answer on this.
My non-expert advice - if he must continue playing during this "off-season", have him throw once mid-week. This allows at least a degree of rest that he would not otherwise get and will provide some degree of maintenance for the in-season arm strenght he has built up.