Headphones are not required for recording voiceover, or for basic editing. You DO need to listen carefully to your setup from time to time to ensure you have great quality, no hiss or hum or room echo, etc.
AKG K240STUDIO semi-open back headphones with replaceable cord are among the best reviewed (reviews: Sweetwater • Guitarcenter • Amazon). Since most studios don’t offer semi-open back headphones, having a pair to take with you can be a good idea.
Many times at a studio you will only be able to hear the producers on the talkback coming through your headphones. Similarly if producers want to connect to your home studio—headphones are better than piping them in through a studio monitor (professional room speaker).
Alternatively, check out the latest reviews of good closed-back headphones under $150 at the WireCutter (Sony MDR-7506’s are a frequent winner) if you prefer using standard closed back headphones when recording in your home studio. The Beyerdynamic DT 770 PRO is always a good choice in closed back headphones as well.
Whether you edit through earbuds (fine for most new voiceover talent) or a headset (don’t edit through speakers), you should to do a hearing test (and room test if you will be listening through speakers), and install and adjust an equalizer on your equipment (on your computer for most people) to “fix” your hearing.
Although reducing breath, plosive and sibilance sounds does NOT require super high-fidelity playback, you DO need to hear frequency range somewhat accurately, so do the hearing test every few years. Your hearing changes as you get older.