I would answer no, but am not sure if I'm missing something and hope you can help me out:
The derivative of a max-pooling layer in a ConvNet is one w.r.t. the maximum value and zero for all others. A max-pooling layer thus propagates the derivative only to the maximum values. Therefore it sounds like a problem in terms of vanishing gradients similar to the dead ReLU?
On the other hand, a max-pooling layer is acting on the activation maps and we want to update the parameters of the filters generated these activation maps, so the max-pooling layer shouldn't affect the gradient w.r.t. to the filter parameters, right? And thus does not reduce the gradients?
Thank you very much in advance!