Yes. Ice dams can peel back shingles, damage flashing, warp fascia boards, fill and tear off gutters, and force water into the roof deck where it causes rot. Each freeze-thaw cycle compounds the damage, so a dam left in place gets more expensive by the week.
The damage isn't only on the roof. Once water backs up under the shingles it tracks into the attic and down through ceilings and walls, ruining insulation and drywall. That's why we create drainage channels first when we remove a dam — relieving the trapped water fast limits the interior damage while we clear the rest.