I think it's safe to say there's no situation when one needs an unbiased estimator; for example, if $\mu = 1$ and we have $E[\hat \mu] = \mu + \epsilon$, there has got be an $\epsilon$ small enough that you cannot possibly care.
With that said, I think it's important to see unbiased estimators as more of the limit of something that is good. All else remaining the same, less bias is better. And there are plenty of consistent estimators in which the bias is so high in moderate samples that the estimator is greatly impacted. For example, in most maximum likelihood estimators, the estimate of variance components is often downward biased. In the cases of prediction intervals, for example, this can be a really big problem in the face of over fitting.
In short, I would extremely hard pressed to find a situation in which truly unbiased estimates are needed. However, it's quite easy to come up with problems in which the bias of an estimator is the crucial problem.