If you add noise to the test set's dependent variables, then your predictions of the noisy (noisier) DVs will be worse, so your MSE will go up. After all, you cannot predict the random noise - that's why it is random noise.
Here is a thought experiment: suppose you have a perfect model and can predict the test set's DV perfectly. Your MSE is zero. Now you add noise to the test set's DVs. It doesn't make sense to change your predictions (since, as above, you can't predict your noise), so all that happens is that the DVs get perturbed. Before, you had perfect predictions. Now, they are not perfect any more. Your MSE increases from zero to some larger number.
And since the MSE is roughly the same as the variance of a normally distributed variable, your MSE should increase roughly by the variance of the noise you are adding.