Do the souls in Purgatory suffer?

Yes. The Church teaches that the souls in Purgatory undergo real suffering. The nature of this suffering is debated among theologians. The Western tradition has generally spoken of a purifying fire; the Eastern tradition has often preferred to speak of a process of maturation and growth (see Benedict XVI, Spe Salvi 48). The defined doctrine requires belief in the reality of purgatorial purification and its attendant suffering, but does not mandate a particular theory about the nature of the fire. What the tradition commonly holds is that the greatest suffering of the holy souls is the pain of separation from God — the aching desire to see Him face to face, delayed by the stain of imperfection that still clings to the soul.