I estimated a CFA in R using missing = "ML" in model syntax. Does this "ML" do something about the no-response bias? Or no-response bias is a totally different thing than dealing with missing observation in a survey study?
You didn't specify which R library you're using to execute the CFA (confirmatory factor analysis). But, for any R library/package, you can always learn about the features and functions by typing "help(package name)"
For example, if you were using lavaan, the command would be help(lavaan)
most (if not all) CFA packages should be able to handle non-response via maximum likelihood estimation or multiple imputation. This does depend on the missingness pattern to some extent (see the paper cited below).
I would not recommend deleting cases, as that reduces power, wastes information, and might introduce bias.