Late to the party, so no idea whether anyone will read this, but...
I think it is useful to keep in mind that imputation will always make up pseudo-information that isn't real information, and as such imputation is a bad thing. Don't do it unless you have good reasons!
When it comes to imputing covariates, however, there are very good reasons for doing it. Think about how much information you lose, and potential biases, when not imputing, and instead doing an analysis on complete cases only.
What imputation does in such a situation is that it makes available real information for the analysis that wouldn't be available otherwise. So even though the imputation as such is bad, it will pay off if the overall effect is positive from involving much more information in the analysis (non-missing covariate values from observations that have a missing value somewhere) than without imputation.
We should never think of an imputed value as "correct" though. An imputed value is a problematic device to achieve something good. Imputation itself adds uncertainty, for which reason multiple imputation is recommended, which basically explores, based on a range of seemingly "realistic" imputation values, how much uncertainty comes from the imputation. (We should also have in mind that the real uncertainty is even larger, because the imputation model itself is uncertain.)
The problem with imputing the response is that, as long as our aim only is to predict the response from the covariates, observations without response carry no direct information about this, and imputation does not change that. Response imputation may lead to underestimation of uncertainty by treating a response as known that actually isn't known, and not even its real uncertainty is known, despite using multiple imputation.
But one can still say that to some extent real information is made available by imputing the response. This concerns the distribution of the covariates, which may have an impact on the model.
In the first case discussed in @Stef van Buuren's answer, all covariates present but response missing, a complete case analysis implicitly uses the covariate distribution of the complete cases, and imputing the response will make the full observed covariate distribution available. To what extent this changes something for the good will depend on what exactly you are doing. It may very well not help or even do harm, in case the imputation model is wrong.
Also if we're running some kind of regression, the regression model predicts the response anyway, and the imputation model needs to somehow improve on the regression model for making sense of imputing the response, in which case we may wonder whether our specific regression model is good in the first place.
A major issue in missing value imputation is that whether (and how exactly) a model assuming MCAR, MAR, or MNAR is correct is strictly not observable, because it critically depends on the values that are actually missing. Unless there is strong background information about the missingness process, we can never rule out with any confidence that the situation is MNAR, and potentially even "evil MNAR" in the sense that any imputation model we try may be quite off.
By imputation we make stuff up, and there is no guarantee whatsoever that we do it well. Van Buuren's point is that if we do it well, it may help, which is fair enough, but not only is there no guarantee, there is an essential barrier to information that can tell us whether we did it well, at least in the sense of comparing the imputed with the true values. What may be possible is that comparing various prediction models on test data, we may find out that a model involving imputing the response on training data may do better predicting the test data than competing models not imputing responses. That'd be a valid justification, but I expect this to happen rather rarely.
Van Buuren is also right that an imputed response may help to impute covariates in case they are missing. However we also need to be very careful about his, because once more, our aim is to estimate the relation between covariates and response, and if we impute covariates by help of the response, we're basically using response information twice whereas the final analysis will treat the data as if the response was only used as response. Once more this may induce an underestimation of uncertainty, but also once more there may be situations in which the risk that this goes somewhat wrong looks acceptable given what we win by doing it, in terms of making more real information available for the analysis. (Personally I hardly ever impute responses, and for doing it I'd want to see that very little imputation involves a very clear improvement in terms of making real information available.)
So the baseline is that imputation never creates real information, and is in itself never good. It becomes good only to the extent that it allows us to involve other real information in the analysis that wouldn't otherwise be available (or only in a much weaker way). This may include use of an imputation model that uses background information that otherwise wouldn't be used. The positive case for imputation is much clearer in most situations for imputing covariates rather than the response. Furthermore imputation should be multiple because single imputation can't be trusted, and we'd like to assess the uncertainty in the imputation process, but keep in mind that there is additional uncertainty through untestable assumptions.
