A fixed-term lease specifies the minimum amount of time that the tenant will rent the property. Despite the tenant's death, the tenancy continues until the end of the lease term. You must then contact the tenant's estate executor or administrator for any issues relating to the tenancy. If there is no estate administrator or executor and nobody has taken possession of the rental unit, you can start eviction proceedings according to your state laws to collect unpaid rent.
In a month-to-month lease, there is no minimum term that the tenant has to rent the property. In normal situations, the tenant only has to give sufficient notice to end the tenancy. With a month-to-month lease, the notice of the tenant's death effectively ends the tenancy at the end of the payment period. For example, if the tenant last paid the rent on the 1st of the month and died on the 18th of the month, then the tenancy will end on the 30th of the month.
Although the tenancy continues with a fixed-term lease, you may have to attempt to find a new tenant to replace the deceased tenant. When the new tenant begins paying rent, the deceased tenant's estate does not have to pay rent anymore. In some cases, the deceased tenant's friend or relative may request to assume the lease and live in the property. You should evaluate the person as you would any new tenant and only rent to her if she meets your requirements.
If you allow a person to enter the deceased tenant's rental unit, you may expose yourself to liability if that person takes any of the deceased tenant's personal property. Even the deceased tenant's children and relatives may not have the authority to remove property from the unit. As such, you should only allow the deceased tenant's executor or administrator to enter the unit and handle the deceased tenant's personal property.