In general, the reason we use non-parametric analyses is that we don't want to have to make (or count on) any distributional assumptions. In other words, you might choose the Mann-Whitney U-test because you think the variances are unequal, for example. In short, non-parametric analyses do not require equal variance.
Let me speculate on what may have been the source of confusion. The Mann-Whitney U-test is a test of one distribution is stochastically larger. That means that if you drew a single value from each population, it is likely that the value from the one distribution will be higher than the other value. Now if the two distributions are the same, except that one is shifted up relative to the other, then stochastically larger implies the higher population has a higher mean. This is only true if the distributions are identical except for shifted. Note further that identical distributions entails equal variances.
If your goals would only be satisfied by a test of the equality of means, and your distributions are neither sufficiently normal nor identically shaped, you could try bootstrapping.