Multiple imputation (MI) is like an unsupervised learning procedure. MI can be used to impute the outcome because imputation doesn't even know or care which variable is an outcome. Rather, you condition on the whole dataset, using every bit of information you can recover to provide better simulations of the missing data's values. Theoretically unbiased and efficient analyses are gained by imputing all missing data where appropriate.
The assumptions for imputing the outcome are the same as the assumption for imputing other variables: the missing values cannot depend on unobserved information. It just so happens that this is more often the case for the outcome than it is for other variables. This is because the collection of the outcome is the primary assessment of the feasibility of the study, but the collection of other covariate data is treated as somewhat incidental. In practice we tend to see that imputing the outcome is a bad idea.
Take as an example modeling diabetes control in a random sample of rural residing elders. Annual household income might be collected by survey, and HBA1c records taken from community hospitals (based on memorandums of understanding and IRB review and blah blah). Now, some folks don't report household income because they simply don't know: they are working part time hourly, receiving social security, and have some revenue from stocks. Impute it. Missing lab records? Well that's important. Did they develop ESRD as a complication from diabetes and transferred care to a community hospital by the dialysis center? That is very informative missingness, and nothing in the data will tell you that. Imputing HBA1c attenuates associations and leads to optimistic predictions of trends.