When to Perform non-linear dimension reduction
The question is too general to have a specific recipe as an answer. Depends on the data and the relation between features but to the best of my knowledge:

When the relation between your features are non-linear i.e. when a non-linear interpretation of correlation between features makes more sense than the linear interpretation.

A detailed explanation of what I mean can be found in this nice answer. Please note that the $X^TX$ in the 4th line of the first paragraph should be $XX^T$, because in data mining, we conventionally use $N\times D$ format for data matrices. So in this example, we are looking for the gram matrix of the rows and not columns.

Hope it helps :)