Those whose devices don't meet the bar are seeing this message, according to BetaWiki (via MSPowerUser): "Your PC does not meet the minimum hardware requirements for Windows 11," says Microsoft's warning. "Your device is not eligible to join the Windows Insider Program on Windows 11. Please install Windows 10 to participate in the Windows Insider Program in the Release Preview Channel."
According to reports, Insiders in the Dev and Beta channels are receiving this message as of this week. David Nation, who has been testing Windows 11 as an Insider on an ineligible PC up until this point, told me that after he did a clean install last night and clicked on the Insider option, he saw a message that has PC didn't qualify. He said there wasn't a link for him to sign into the Insider program at all.
Microsoft's minimum requirements for Windows 11 have been unclear from the start and continue to be a moving target. Officials have said they are disqualifying a number of not-very-old PCs from being eligible to get Windows 11 due to security and reliability concerns.
Last week, Microsoft added a few processors to the "qualified" list. But they also created a loophole via which those running PCs which Microsoft deem unqualified to get the Windows 11 release and run in an "unsupported state." Unsupported in this case means these PCs may or may not receive security and driver updates via Windows Update.
Update (September 2): Microsoft is now emailing Insiders whose machines are deemed ineligible for Windows 11 to let them know they need to install back to Windows 10 and rejoin the Release Preview Channel (which is testing Windows 10) if they want to continue to be a Windows Insider tester.
The note (a copy of which @mniehaus) posted on Twitter and which I've included below) says users on these machines "will not receive new Windows 11 Insider Preview builds from the Dev Channel going forward." The note doesn't say anything about those in the Beta Channel ring who are testing Windows 11. The note also doesn't say what will happen to people who don't clean install back to Windows 10 and decide to keep running the Windows 11 builds they have.
Update No. 2 (September 2): Microsoft officials used today's new Insider build blog post to attempt to clarify a bit more which Insiders are going to be booted off Windows 11. It sounds like it may be a smaller subset of individuals than originally thought. More details here.