Why are correlation consistent basis sets used with DFT calculations?
It is generally recommended not to use a cc basis set with a DFT method (and I guess conversely, a basis set aimed at DFT should not be used with a coupled cluster method).


This statement glosses over some specifics that might be important.


There is nothing technically wrong with using correlation-consistent or ANO basis sets with DFT, unless the basis is deficient to begin with (cc-pVDZ). That is, using cc-pVQZ to optimize the geometry of a small organic molecule will not lead to more "incorrect" results than 6-31G* unless linear dependencies arise in the basis due to over-completeness.
However, it is most certainly inefficient to use these basis sets for general usage (SCF-type methods) due to using general contraction rather than segmented contraction. Most modern integral algorithms still have a difficult time dealing with highly-contracted basis sets. This is one reason to use the TURBOMOLE/Ahlrichs/Karlsruhe Def2 and Frank Jensen's segmented polarization-consistent (pcseg-n) basis sets.
"Using a basis set aimed at DFT should not be used with a coupled cluster method" is more tricky. The TURBOMOLE basis sets are certainly used for MP2 and coupled cluster calculations, however they are more appropriate for "single" calculations (not CBS extrapolation), and only with the largest basis one can afford. They were designed to capture the total mean-field energy, which decays exponentially, not the correlation energy, which decays like ~$(L+1)^{-3}$, where $L$ is the maximum angular momentum being considered. The correlation-consistent and atomic natural orbital basis sets are better for coupled cluster because they were designed from the beginning to recover the correlation energy.
The use of Pople-style (STO-nG, 3-21G, 6-31G*, 6-311++G(d,p), etc.) basis sets would be incorrect with wavefunction-based methods, perhaps even for qualitative understanding depending on how "difficult" your system is. Like the TURBOMOLE basis sets, they are designed to capture the mean-field atomic energies, but many have constraints (identical s- and p-type exponents, the "SP" functions that are often seen) due to the computational resources available at the time of their creation. Many of the older basis sets (notably 6-31G and not 6-311G) were optimized to use Cartesian rather than pure d-type functions, leading to potential inconsistencies in results. It is also not as obvious how to systematically improve the quality of the basis sets, and none are larger than triple-$\zeta$. Ultimately, the virtual MO space created by these basis sets is not sufficient to recover the correlation energy, and the benefits of using coupled cluster are completely lost.