Keras functional api explanation of activation() layer?
As stated in the docs, the activation layer in keras is equivalent to a dense layer with the same activation passed as an argument. 

This would be equivalent

x = Dense(64)(x)
x = Activation('relu')(x)


is equivalent to

x = Dense(8, activation='relu')(x)




As per your example if the activation layer is used as a layer, this will act as a transformation of the outputs of the previous layer. 

As you can see in the model

inputs = Input(shape=(784,))
x = Dense(32, activation='tanh')(inputs)
x = Activation('relu')(x)
predictions = Dense(8, activation='softmax')(x)


The output of the first layer is the result of a densely connected layer with a tanh function. Then these outputs will each be transformed by a relu function. You can see how this is simply a transformation and does not introduce any new model parameters using 

model.summary()