What does is mean when the standard deviation is 0?

Intuitively, a standard deviation is a change from the expected value.

For the question you asked, this means that the change in the "results" doesn't exist, which doesn't really happen. If the standard deviation is 0, then it's impossible to perform the test! This shows that it's impossible to compute the probability with the "null" standard deviation from this form:

z = (x - µ)/σ

If σ = 0, then the probability doesn't exist.