Why should we use (or not) dropout on the input layer?
Why not, because the risks outweigh the benefits.

It might work in images, where loss of pixels / voxels could be somewhat "reconstructed" by other layers, also pixel/voxel loss is somewhat common in image processing. But if you use it on other problems like NLP or tabular data, dropping columns of data randomly won't improve performance and you will risk losing important information randomly. It's like running a lottery to throw away data and hope other layers can reconstruct the data.

In the case of NLP you might be throwing away important key words or in the case of tabular data, you might be throwing away data that cannot be replicated anyway else, like gens in a genome, numeric or factors in a table, etc.

I guess this could work if you are using an input-dropout-hidden layer model as you described as part of a larger ensemble though, so that the model focuses on other, less evident features of the data. However, in theory, this is already achieved by dropout after hidden layers.