The successive convolutional layers in popular CNN architectures learn representations from very general (edges) to very specific (dog breeds) from the input to the last conv. layer respectively. Depending on how different your new domain is will dictate on what layer you will take the CNN representation from (early layers or later layers) and whether a generic ImageNet model will work at all.

Of course if your new domain is very different than the distribution of the original training data, it is a good technique to fine-tune the network a bit with your data.