What rights are there for a subcontractor if they do not file a notice of furnishing on a private commercial/industrial project?
The answer depends on whether the owner or general contractor properly filed a notice of commencement pursuant to R.C. 1311.04. If the notice of commencement was properly recorded, then the failure of a subcontractor to serve a notice of furnishing is fatal to a subcontractor's lien rights.
If no notice of commencement was recorded, then subcontractors are not required to serve a notice of furnishing in order to preserve lien rights, and the normal lien timeframes apply.
A notice of furnshing is also not required for anyone under contract with the owner directly. But since your question implies that you are a subcontractor under contract with the general contractor, this exception likely does not apply.
A Notice of Furnishing is required for commercial construction projects only when a Notice of Commencement has been recorded. So if a Notice of Furnishing was not required, then the subcontractor might still have lien rights. If a Notice of Furnishing was required (and the subcontractor did not serve one), the subcontractor can still serve the Notice of Furnishing, which would give that sub lien rights for work performed in the preceding 21 days. In any event, subs always have the right to sue the contractor for breach of contract (subject to any pay-if-paid provisions) and may have claims against the owner for unjust enrichment.