If I hold non-residential property on lease from a third party that in turn holds the property on temporary emphyteusis from Lands Authority, what will happen when this emphyteusis expires?

As the sitting tenant, you may (at Lands Authority discretion) be granted a new lease as long as you are not the lineal descendant or ascendant of the emphyteuta (or spouse thereof) and as long as the lease originated at least 10 years prior to the expiry of the emphyteusis. This lease will be a direct one, without the need for tenders but at a new rent established by Lands Authority. If, upon expiry of the temporary emphyteusis, you would have held the lease for less than 10 years, or you are otherwise a tenant that is a lineal ascendant or descendant of the emphyteuta (or a spouse thereof) then reletting can only be carried out through tenders, but as sitting tenant, you will be granted the right of first refusal. For further details see Clauses 18.1 and 18.2 of the Policy for the Disposal of Non-Residential Premises.