Are there ILS approaches where a non-precision procedure isn't allowed?
Yes, there are.

Note: This is as opposed to those ILS approaches listed as ILS or LOC in the title—such as KGEG ILS or LOC RWY 3—which do have an authorized LOC approach.

Special Authorization Cat I ILS, and Cat II &amp; III ILS approaches do not have authorized LOC procedures. These are precision only approaches.

See KGEG approaches, for example. Specifically, the ILS RWY 3 (SA CAT I), and the ILS RWY 3 (CAT II &amp; III) approaches do not have authorized LOC procedures.

See also this screen cap of the KGEG ILS RWY 21 (SA CAT I) approach:  

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