As a trustee handling other people's money, you must have an investment plan that complies with both the trust document and state law. This is the "prudent investor rule."
Being a trustee can involve a lot of work. Depending on the type of trust—living trust vs. long-term trust—there may be ongoing responsibility, so compensation is usually part of the deal.
If you are named as a POD beneficiary on the account, or if you're managing the estate in a probate proceeding, closing the checking account and getting the funds should be fairly easy.