Remove Bootstrap's validation icons

If you are compiling BootStrap's SCSS you can include this before you import bootstrap.scss:

$enable-validation-icons: false;

You can simply remove the icon by by setting the background property to none for that specific CSS class/pseudo-class

.was-validated .custom-select:valid {
  background-image: none;
}


.was-validated .custom-select:invalid {
  background-image: none;
}

If you would like to remove the validation icons, but retain the arrow icons on the select input, it can be achieved by setting the background to the following

.was-validated .custom-select:invalid {
     background: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 4 5'%3e%3cpath fill='%23343a40' d='M2 0L0 2h4zm0 5L0 3h4z'/%3e%3c/svg%3e") no-repeat right .75rem center/8px 10px;
}

Update for bootstrap 4:

Setting the background image to none for the particular selector retained the background color and the dropdown button.

.form-control.is-invalid{
    background-image: none;
}

Update for those working with Bootstrap's SASS/SCSS:

Please refer to PCalouche's excellent answer!