Yes. yes they are
They’re definitely important. The intent of a headshot is to show people you. Part of communicating who you are is your clothes. Are you a business person in a field that has a high standard on how someone dresses, such as suits and ties? Are you in a creative field where it is important to show your sense of fashion in your headshot? These are the types of questions that you need to consider. We only shoot from the chest up, but viewers of your headshot will see a good portion of clothing that is communicating something about you.
keep it simple
My general rule is that simple is better. Keep colors muted and without patterns. Choosing neutrals and grays is best or least not bright or fluorescent colors. Even if you are the most eclectic fashionista I believe that you can communicate your style without the viewer being distracted and/or put off by your clothes.
Remember hair style communicates your style loudly as well. See my post about makeup and hair HERE.
let's make your headshots look like you
Most likely people will see your headshot before they meet you in person. If you show up to that first meeting and you look nothing like the headshot then that's a problem. If I were hiring someone and their headshot was drastically different from how they look in person my instinct might lean toward distrusting them. Granted it’s a subjective area, but it’s something to consider. My goal for a headshot is showing people you in an accurate yet flattering way.