Not always. Doing a good job requires persistence and attention to tedious detail, but not necessarily a law degree. If assets must go through probate court, the process is mainly paperwork. In the vast majority of cases, there are no disputes that require a decision by a judge and the executor may never see the inside of a courtroom. It may even be possible to do everything by mail. (To learn more about the duties of an executor, see Nolo's article What Does an Executor Do?)
An executor can probably handle the paperwork without a lawyer if he or she is the main beneficiary, the deceased person's property consists of common kinds of assets (house, bank accounts, insurance), the will seems straightforward, and good self-help materials are at hand. If, however, the estate has many types of property, significant tax liability, or potential disputes among inheritors, an executor may want some help.
One good resource is The Executor's Guide, by Mary Randolph (Nolo), which guides executors through the process of winding up a loved one's estate, step by step.