When the standard error of the difference is 0 then you might go with a non-parameteric test. A sign test would be good. Rather than give you the probability on a t-distribution, it gives you the probability of that many successes (differences in the same direction). That would typically be a meaningful and useful kind of p-value to describe.
The variance of 0 means that all of the differences were the same. This can happen for a variety of reasons, for example, insensitive measurement, or genuinely extremely low variance of the effect. Perhaps you even rounded off the variability. It's very rare though, if you have data that follow the assumption of the t-test. Your data probably require a non-parametric test in the first place.