Will a dataset with multiple labels perform better than with binary labels? Suppose I have a dataset comprised of garbages. Will a model perform better if I only label the dataset with biodegradable or non-biodegradable?
Or will it be better if I label them with plastics, paper, cardboard, glass, and organic?
In fact, I also plan to further increase the number of labels for example, the plastic label with be comprised of a large number of brands of plastic wrappers, etc.

I think that having a large number of labels is detrimental to computational performance in both training and evaluation since for example, Linear Discriminant Analysis will not lead to much reduced dimensionality due to the nature of the subspace spanned by the centroids on each label.
Neural networks would have a very wide top softmax layer and I am pretty sure that would require a wider architecture.
Aside from model issues, I could easily suffer from class imbalances. Is there any merit to having multiple labels ?

If the data is clustered in the way you describe in your comment, then for the two-class problem you are likely to have a non-linear decision boundary. As linear models can't model non-linear decision boundaries, they are unlikely to perform well here. However, an appropriately sized non-linear model such as a neural network should be able to model this (for more on linear vs non-linear models see Paul Fornia's answer to "Linear vs Nonlinear Machine Learning Algorithms"). If the NN is wide and/or deep enough it will effectively learn the cluster groupings during training.
Correction (in response to @kjetil's comment): You could still use a linear model if the features can be transformed to a higher-dimensional space in which the decision boundary is linear (e.g. by using feature interactions or squaring, cubing etc features).
Transforming your data to a multi-class problem may give you multiple linear decision boundaries, which should help improve a linear model, if for some reason you are restricted to using linear models.
Your point on class imbalances prompted another thought. By separating the classes into sub-classes you may lose information that is common between the subclasses. For instance, consider classes of 'hard plastic' and 'soft plastic', which probably have similar values for some attributes. If they are separate classes the model may treat these shared attributes as "not useful" and ignore them. Conversely the model may place too much importance on differences between 'hard plastic' and 'soft plastic' which are not useful for bio/non-bio classification.
