For asynchronous interviews, it's not essential to wear headphones during recording, but they are recommended.
Headphones can be useful for hearing the quality of your recorded audio, allowing you to make adjustments to your audio environment. If your voice is faint or muffled, or there is some echo in the room you are recording in, this will be much easier to detect using headphones.
It's just as important to prepare your environment for recording by eliminating any external sources of noise.
Note: In traditional (synchronous) interviews, your microphone can pickup and record the other person's voice coming out of your speakers, so it's often recommended to wear closed-back headphones. Asynchronous interviews avoid this issue.