The veil Paul speaks of is the carnality of a deceived mind formed and shaped in the world's Satan-manipulated cultures. It is so antagonistic to the true God and His Word (Romans 8:7) that it fights the very approach of God to heal them through a great, freely given gift, just as the first-century Jews opposed and rejected Christ.
However, be aware that the miraculous removal of this veil of blindness by God, through the wonderful gifts of His Spirit and of a great hope, also places an obligation on us. With the blindness gone, we are granted the ability to choose between God's way and the world's way for the first time in our lives. Choosing to submit to God provides our witness of God, as well as being the means of building the character God greatly desires to create within us.
However, the effects of the self-centered way of life we have absorbed through the course of this world remain in our attitudes and characters, becoming what must be overcome. It will dog us all our converted lives as a means of testing our determination to be in God's Kingdom, as well as our love and loyalty to our God and Savior.