Unfortunately, probably not. I would generally just tell you to file for probate on your own without hiring an attorney in order to save on the expenses, but you will be unable to do this in Tennessee. While Tennessee does not require that you hire an attorney to probate the Will, anyone who is a named fiduciary of the Estate is not allowed to represent themselves in their capacity as a fiduciary. As your father's Will names you as the Executor, you would be required to have an attorney file the petition to probate the estate. On the plus side however, the Executor is entitled to reimbursement for out of pocket expenses that are required to administer the Estate, so if there's any money in the Estate at all, you can probably get your $714 back. The only other alternative I can think of would be if your father had a Financial Power of Attorney appointing you (or your mother) as his Attorney in Fact. If that were the case, you would have the power to access the safe deposit box without the Will. I am sorry it's not an inexpensive answer!
*Please note that I am not licensed in Tennessee. My answer is based entirely on research of applicable Tennessee probate laws.
The answer you received from the court is correct. They will not issue Letters Testamentary without opening an estate administration. From what you say, it sounds like a small estate probate will be possible. Although the court will not allow an executor to represent himself in a regular probate, they might allow you to do so for a small estate matter. You might also try telling the bank that the estate is not worth probating so you do not plan on opening an administration. In that event, ask the bank how will they handle property sitting in a safe deposit box in your father's name. Also, given that the monthly safe deposit box fees will not be paid, what will they do to close out the safe deposit account. You might be able to force their hand on it.
I would add the following to the previous answer. Some jurisdictions allow for the use of a small estate affidavit in order to avoid probate. When this is available, the estate is generally under a benchmark of value ($15K to $100K depending on the jurisdiction). The small estate affidavit may be available on line.
In addition, some jurisdictions have a truncated form of probate to open a safe deposit box. You may want to look into your local law on this.
I am not licensed in Tennessee and can not tell you if this option is open to you in Tennessee. I would not let the box go.
Disclaimer: This response is for informational purposes only and does not constitute legal advice or a substitute for legal counsel. This response is formed based upon the facts given as interpreted. Unknown or variable facts could result in a different response. This attorney is licensed to practice law solely in the State of Illinois. Online readers should not take any action based on this response without first obtaining the advice of professional legal counsel licensed within your state. The response does not create an ongoing duty to respond to questions nor does it form an attorney-client relationship.