It would cause damage. The faster you go, the worse is the damage. Even just the cosmic microwave background will eventually turn into dangerous gamma radiation

It would cause damage. The faster you go, the worse is the damage. Even just the cosmic microwave background will eventually turn into dangerous gamma radiation, so you will need heavy lead shields in the front, which makes traveling even more expensive and difficult. At certain speeds, you should think of your spaceship as a drill penetrating the inter galactic medium. And even then the journey will take millions or billions of years (in the reference frame of those staying at home).