I've done quite a bit of debugging on this...
My FORM tag didn't quite begin just inside the BODY tag, and didn't quite end just inside the closing BODY tag. There was a bit of stuff between the opening BODY and opening FORM and between the closing FORM and closing FORM.
I moved the opening and closing FORM tags outboard, so they were essentially just inside both the opening and closing BODY tags and clearly encompassed everything on the page.
Now, I no longer get the error...but OPE doesn't work either.
Essentially, the page reloads with the OPE toolbar at the top, but there are no WYSIWYG buttons loaded inside it, and none of the areas that are supposed to be editable are highlighted in the purple outline. Clicking on them does nothing. The page loads up as if nothing were editable.
On a v6 site after an R2 upgrade, any attempt to do On-page Edit results in:
Control 'ctl02_ctl00_HeaderContentContainer' of type 'OnPageEditInputControl' must be placed inside a form tag with runat=server.
This error occurs after right-clicking and selecting "On-page Edit" -- you don't even get a chance to see the editing interface.
The control IS in an server-side FORM tag. It worked fine under R1, and none of the templating has been changed in the upgrade.