Kind of the whole idea behind DOI is long-term persistence. It is a way to permanently identify a specific object, (hopefully) even decades after its creation and to be able to still look it up, no matter how often e.g. publishers change their website layout.
What you are trying to do would kind of produce the orthogonal problem, in that you have a permanent identifier pointing to an object that is continuously in flux. While I don't think that this is technically forbidden, and indeed there is sometimes a grey area in regards to later revisions of a document, I would still call it a bad idea.
Let's say that you are able to obtain a fixed DOI for your dissertation early on while it still is changing. Then that DOI is of no help in referencing it, as that reference might refer to content that is not there anymore in the next version. Any automated systems that index it might also index the first version and then never bother to update. You might be able to change the metadata associated to the DOI accordingly, which might even force updates down along the line, but that sounds like a lot of work for no benefit.
Two other remarks:
If you are using some sort of version control system, this can be used to reference a specific version of your work. However a DOI pointing to the repository with all the versions is not the same as a DOI pointing to the resulting text.
While "continuous delivery" could indeed be a useful approach to writing a dissertation, you should not necessarily take that literally. There is no need to deliver every early draft to the whole world. If you are lucky, some people (other than those who grade it), will be interested in reading it, but they certainly will not be interested in reading it again and again in a slightly modified form every two weeks. Instead they will be perfectly happy to read it once it is finished.
