Do the test dataset feature order and inference (real world) feature order have to be the same as the training dataset? For example, if features are in the order (a,c,b,e,d) for the training dataset, does that particular order have to match for the inference and test dataset?
Generally, order matters. A (trained) Neural Network (NN) is just a mathematical function trained on taking some given input and producing the corresponding output. So, if you train a certain node on producing large output if (and only if) an animal is present in a picture (for example), but later you give it the numeric evidence for a car being present in the image, it will still produce large output, indicating an animal being present in the image. This is simply because a node doesn't know what it is receiving or supposed to detect. It just follows its standard mathematical procedure.
So, if you train your network on one kind of input data, you must provide the same kind of input data during testing (or at least you cannot simply exchange inputs or outputs of certain nodes without manually correcting for that in the remainder of the network).
In the most simple case, consider whether you could simply swap the inputs to that simple function: $f(x,y) = x^2 + y$. If you swap the inputs, the output will be different. The exactly same applies to NNs.
Addition: I think this post explains and especially illustrates the intuition nicely.