By programmable, I suppose you mean that it contains information or can be altered in response to some input or stimulus. The answer is "no" for both. Well, sort of.
Does noncoding DNA contain information? By definition, no. There are probably many regions of the genome that appear to have no information, only later to be found to contain introns, regulatory elements such as enhancers, boundary element, MAR/SARs, targeting sites, etc. Even functional tests (such as removing the region) may not reveal anything because the effects could be minor, or only evident under special conditions. But arguably, if you remove a region and it has an effect on the organism, then it's not really a noncoding DNA, it's just you didn't see the coding before hand.
As for the latter, can it altered, the answer is again "no," or at least "apparently not." Intergenic regions (those stretches of DNA that do not contain obvious or characterized transcribed regions or their control elements) are very stable between organisms and even between species. They seem to have a mutation rate expected for having no information, and thus free to mutate slowly without being swept away. There is no evidence (as far as I know) of any region of the genome being purposefully altered, with the exception of a handful of specific genes whose regulation is controlled by DNA nicking or some such.
Perhaps I am missing your question, being a biologist and not really knowing what a "Turing Machine" is. If I misunderstood, please clarify.