There comes a point where Cross Fit (or enter any activity) can become chronic; and, over time, put undue stress on the body. However, this aside, it's important to let your muscles repair properly; and know when to take a break.
To address your question, directly, I'm guessing there's a danger to not keeping your muscle fibers active, over a longer period of time. But, as I previously wrote, muscles will need time to repair themselves.
While you're healing up, think of what little strength movements you can do, given your situation. I personally haven't done Cross Fit very frequently. But, having just completed one session - this past Saturday, after having not done any in months (forgot exactly how long ago was my last workout), I found I wasn't in the shape I used to be. On the flip side, I found I've recovered better (a nice post workout meal helps in this regard), than in the past. Granted, I've been doing more play (martial arts) than actual Cross Fit. I suppose, there was a little give/take, in the process.
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