It is difficult to give you exact %'s. In general, most body MRI should be performed both with and without contrast, unless there are contraindications. This is important for characterizing masses in most organs, but is especially important for characterizing liver masses which display different enhancement patterns. It is not just whether or not something enhances but the way in which enhances. For example, a benign hemangioma typically begins with peripheral discontinuous nodular enhancement with gradual central fill in on delayed images. On the other hand, a malignant hepatocellular carcinoma typically demonstrates initial hyperenhancement followed by subsequent washout on delayed images.
Also, MRI is not really used so much for initial screening of tumors. Typically, other tests such as US and/or CT are performed first. Then, if there is an indeterminate lesion, MRI is performed to further characterize the lesion.