Do the weights of a neural network get any kind of IP protection?

Depending on the wording of the claim as a whole it is possible but any claim with the specifics of weights and, correspondingly, a specific architecture of nodes would be easy to get around. As an example of language in a claim of an issued patent -

a trained generalized regression neural network trained using test reservoir data and test relative permeability values, with the trained generalized regression neural network for processing the actual reservoir data to determine a relative permeability prediction of an actual relative permeability in the hydrocarbon reservoir from the actual reservoir data; and

It is specific as to application and the source of the data but not to the gritty details of the neural network. One could imagine a dependent claim that nailed down the characteristics of an example network to greater and greater detail, including weights. It would not be the weights that gave patentablity.

Do the weights of a neural network get any kind of IP protection?

Almost certainly not, as they are obvious (in a patent sense). The reasoning for this is a little bit indirect.

Neural networks in general are very well-known and commonplace. In addition, training methods for neural networks are very well-known. By applying a known training method to a known neural network, you are doing nothing more than what has been done thousands of times before. The fact that you have a unique series of values as an output is not in itself non-obvious, since they were obtained by well-known and obvious techniques. As such, it would be obvious for the person skilled in the art to try the same approach, and they would then obtain substantially the same results.

Unless you have invented a unique type of neural network, or a unique method for training, or a unique step of filtering data, or the neural network is being applied in a suprising area, then you're really doing nothing non-obvious.

Copyright

Copyright is off-topic here, and I would encourage you to post a similar question on Law.SE to get a full answer.

However, as an aside, almost certainly there would be no copyright in the weights of a neural network, since they would not involve even a modicum of creativity.