The influence of speech on writing is quite well-known, so I won't go into that.
Coming to the influence of writing on speech, Chomsky gave a talk at Google recently, where he spoke about various things. At one point, a questioner asked about the relation between writing and speech. The question was very different from yours, but his answer applies here. After Chomsky talks about the work of Ken Hale on "cultural gaps" (probably this article), Chomsky mentions another study, of Middle English (author and title not mentioned) that showed that "the use of complex constructions... which had embedded elements in them increased as literacy increased". Chomsky then posits an explanation for that, having to do with speech being constrained by our limits of short-term memory. He concludes that "once literacy spreads, you get much more complex linguistic usage, even in speech, because it carries over from writing to speech".
Apart from that, the influence of speech on writing is getting more and more prominent these days because (1) most people are literate and (2) many people communicate a significant portion of day-to-day affairs via short texts of various sorts (instant messaging, SMS, tweets, etc.). The drive for economy has led them to invent a huge number of abbreviations. Some of these have spilled over into speech as abbreviations (OMG!) and words (ASAP => ay-sap, YOLO => yo-lo). I'm sure a teenager could give you many more examples.