No. The price of your headphones has nothing to do with hearing damage. Volume and listening time decide the risk, so a $20 pair played at a sensible level is as safe as a $400 pair.
Cheap headphones can push you into a bad habit, though. A pair with weak noise isolation lets traffic and engine noise in, and most people turn the volume up to drown it out. That’s where the risk comes from. If you listen in loud places, a budget pair with noise cancelling, or at least ear pads that seal well, lets you keep the volume lower.
Keep ear pads and tips clean too, and replace pads once they crack or flatten. A worn pad leaks bass, and that tempts you to turn the volume up again.
