Can you please explain when (under what circumstances) the dimensional reduction and/or feature selection techniques are not effective to mitigate the curse of dimensionality?
This is helpful when the data is not understandable and you don't have any data dictionary or you have too many columns which doesn't make any sense even after through investigation, then it is wise to go for Dimensionality Reduction.
There is very good chance that you loose information at granular level, for example you gave 100 columns and you got 10 PC's(Applied PCA) which can explains most of the Data. As you can only get that much out of that technique.
Feature Selection: If you miss even a single feature which is significant WRT target variable but you overlooked that feature then your model might not be able explain the most, you are even satisfied with the result but still there is window of improvement. You should be very careful and you need check as many times(many iterations) as possible to build a good/decent model and get the best results out of it.
I hope this may help you.
Both approaches are 'lossy'.
For PCA, assuming you retain fewer components than variables, you necessarily throw away information. If you don't have a few components that capture most of the variance, you could throw a lot away.
For feature selection, (I assume you mean an automated approach like lasso), you again throw away information, hopefully only incremental so. But, depending on how you structure the selection routine, you let an algorithm make some level of design decisions which result in less data.
Whether this is bad or not depends on intent. Fire pure classification it's less an issue. For interpretability it can be essential.