If your spouse is the named beneficiary of your HSA, your HSA becomes the HSA of your spouse upon your death, subject to the completion of documents as required by your bank. Your surviving spouse is not required to include any amount in gross income for tax purposes as a result of your death and he or she is subject to income tax only on those distributions that are not made for qualified medical expenses.
If someone other than your spouse is named the beneficiary of your HSA, the HSA will no longer be considered an HSA as of the date of your death. Rather the beneficiary is required to include the fair market value of the HSA assets as of the date of death in his or her gross income for the taxable year that includes the date of death. The included amount is reduced by the amount in the HSA used, within one year of your death, to pay your qualified medical expenses that incurred prior to your death.
If there is no named beneficiary of your HSA, the HSA will no longer be considered an HSA as of the date of your death, and the fair market value of the HSA assets as of the date of death is included in your gross income for the year of your death.