How do I programmatically turn off the wavy red lines in a Microsoft Word document via VBA?

You can mark the current document as already spell-checked:

ActiveDocument.SpellingChecked = True

or you can disable spell checking for a range of text:

ActiveDocument.Range.NoProofing = True

You can also simply disable the display of the red squiggles:

ActiveDocument.ShowSpellingErrors = False

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Note that the same things can be done for grammar-checking (green squiggles); e.g.:

ActiveDocument.GrammarChecked = True

will mark the current document as already grammar-checked, thus effectively disabling it.

ActiveDocument.ShowSpellingErrors = False