No, there's no reason to worry.
I called Discover's customer service line about this and got a great explanation from Alex in Salt Lake City. To paraphrase/summarize:
The website shows that warning because not all Discover cards earn Cashback Bonus in the same way. Your Discover it® card always earns 1% (plus more for rotating bonus categories), but the older Discover More® card rewards program earned only 0.25% for the first $3,000 in purchases every year, then bumped the percentage up (but still excluded certain discount stores like Walmart). Reducing the number of days in the "anniversary year" means that the period during which the More card earned 1% would be reduced due to when the statements close.
So, the warning might not apply to your card type, but the website still shows the warning to avoid even the possibility of affecting your rewards without knowing it.
This was a fascinating bit of historical trivia for me, since I had no idea that older Discover cards had such complicated reward structures. Every card I've ever held (starting in 2008) has had a base reward rate not dependent on spending, so I was unaware of spending tiers and how billing dates might affect them.