I am assuming this property is in Oregon. You may carefully store the tenant's personal belongings. The tenant's family may file a probate, or perhaps not. Either way, you have to abide by the provisions of the landlord tenant act pertaining to a tenant's personal property, which is a very technical area, and has provisions for times when a tenant passes away. I recommend you hire legal counsel to help you.
The tenant's belongings are the property of his estate. If he or she arranged for it while living, an executor will be appointed, who will have the task of taking his property and distributing it according to the will (or to the intestacy statute, if there's no will).
If the tenant instead died intestate and did not make arrangements for handling of their estate, then in theory, the property still should go to the persons designated by the intestacy statute - children, if there are any; parents, if there are not - but your duty to find these people is limited. You should consult with an attorney who can find out if the estate is to be probated. In the meantime, you should, if possible, keep the tenant's things, but you can move them out of the rental space and start renting it again right away.
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