Protein shakes themselves are not a cause of acne, but there is anecdotal evidence to suggest that whey protein powder and other low-fat milk products can increase acne in some people.
Whey protein increases the hormone IGF-1 which encourages muscle growth, but it's also known to influence sebum production. An increase in sebum can create the perfect environment for acne.
It's worth noting, however, that the correlation between protein powder and acne has only been found in whey protein products. If you suffer problems after using whey protein you can switch to a plant-based protein to avoid increasing sebum and increased acne.