I've never heard of this before, and I spend quite a bit of time teaching both InDesign and Acrobat. You can create simple objects (e.g., a rule after an form item), or a square for a checkbox in InDesign. When exported as a PDF file, if you use the Form Field Recognition feature, these can be turned into a text field or a checkbox field.
But there are no controls in InDesign that would affect how a multiline form field (which can only be created in Acrobat) prints out. Adding layers to an InDesign file will have no effect on that.
Have you asked in the Acrobat forum about how to do what you're trying to do?
How text behaves within a text form of a PDF has nothing to do with Indesign layers or Acrobat layers. The properties of the text form field itself controls this. You have these settings: multi-line, scroll long text and limit number of characters. If the form has either multi-line and/or scroll long text selected, then a user will see all the text as it is entered, but once completed, only the text that fits the form field is viewable and printable. And this sounds like what you describe.
I think the solution would be to set a limit to the amount of characters of the text form. This would prevent any unprintable text within the form field.
I agree with Jeffrey's suggestion.