Do you pull off the dead roses from carpet rose bush?

When you want your rose bush to continue to bloom, best practices dictate that you cut the spent bloom from the stalk -- cut at the junction of the bloom stalk and the larger stalk -- so that the energies of the bush can be invested in forming more buds and blooms.

Pulling them off can damage the bush. A small, clean, sharp pruning shear is a better tool.