My big toe really hurts after I wear tight shoes. Is this bad for my feet?

Can be. Tight shoes can lead to ingrown toenails and other deformities like bunions and hammertoes. If the shoes are tight enough and you wear them long enough you can even create some nerve damage in the great toe.
Something that is. Painful is rarely good. Other than switching to shoes that fit better, you may want to look into whether you also have foot problems.