The definition of fast and slow is a bit complicated.
I just tried an organic molecule with about 100 atoms and ran two jobs: One with def2-TZVP (1647 AO basis sets, 1872 CAOs) and one with cc-pVTZ (1938 AOs, 2205 CAOs). Pure Hartree-Fock single-point energy calculation, both jobs needed 13 SCF iterations.
Timings: def2-TZVP 31 minutes, cc-pVTZ 44 minutes
Same job but with RI-DFT using PBE functional:
Timings: def2-TZVP 81 seconds (15 iterations), cc-pVTZ 94 seconds (13 iterations)
So, yes, Dunning basis sets are slower, but for real-life applications the difference is not really grave. Although Susi is of course right that for heavy elements and really large basis sets things are worse...