You mentioned that the property is in Ohio. In Ohio property values for the purpose of the property taxes are reassessed every 6 years (source). It doesn't appear that the change of ownership is a factor, only the passage of time (and changes to the property may trigger reassessment, such as remodeling or alterations).
Concealing the transaction from the auditor may affect such reassessment, but only marginally. It will only affect it significantly if everyone does that, otherwise the auditor will just use comparable sales around you to determine the value of your home.
Keeping the title in the LLC and passing the LLC around may have other drawbacks:
- You may find it hard to get financing for such a transaction, conventional banks don't want to have LLCs as debtors
- You may find it hard to get title insurance, since the title doesn't technically change.
- It is difficult to confirm the LLC owners (in some States even impossible), so you may have potential scams involving LLCs
- You're bound by the existing LLC operating agreement and incorporation papers, which may or may not be convenient to you. You may be limited as to how you can change them.
- You will be bound by the existing LLC contracts, which you may not even be aware of.
I'd suggest to transfer title to yourself or to entity which is in complete control by you (your own LLC for example). You should probably ask an attorney licensed in Ohio both about the real estate aspects and any other potential pitfalls of such a transaction.