What characteristics would describe a "Non-Linear Language"? For instance would it use Logographs (Chinese, Egyptian, etc.)? Or could it use IPA sound symbols? What would make the language Non-Linear in a speech context (not written)?
I'd like to argue that there is no such thing as a non-linear language. Every language needs to convey some some information from the speaker to the listener (or signer to the viewer, or using another channel at all), and that information can by represented as a stream of bits that is linear in time. I have heard of scifi writers talking about "non-linear languages" used by some non-human beings but no one of them provides samples or a tentative description of those non-linear languages.
P.S. My notion of "linear" is not identical to the term "linear language" in Theoretical Computer Science, those "linear languages" are a proper subset of the context-free languages (and from linguistics we know that natural languages are sometimes not context-free). In my answer I just mean "being able to be represented by a linear string of symbols".
I agree with the answer by Sir Cornflakes; to phrase it in a different way: there is plenty of information that is non-linear. Thought, concepts, relationships, images, etc. But language is a code to transmit this information between people. I have an image that I want to describe to you, but I need to linearise my description, because in languages you have a temporal/sequential element in the transmission. Even German and hieroglyphs do that. There is a fixed sequential ordering when you read hieroglyphs (or Hangul for that matter, which is also two-dimensional).
If you want to look at an image, you cannot look at it all at once, so you need to select a sequence in which you visit the various areas of it. That is then linear again. And depending on the order, the image might have a different meaning to you, just like word order in many languages.
So, language being a means of communication, there has to be a linearity to it.
A purely spoken language could not be non-linear, as sounds must be expressed one after another. The only real exception to this that I can think of would be a language that must be spoken by multiple speakers simultaneously.
Written languages have significantly more options though, as they aren't restricted by time. The author will have to write in a certain order, but nothing has to guarantee that the reader reads in that same order. I would recommend reading this essay on nonlinear languages and looking at UNLWS (which I think is an example of a nonlinear language done well).