Your question indicates that you want to use a link to your LinkedIn profile in your CV because "it'll be far more up to date than the CV in my dissertation." Are you suggesting that (a) you are going to include a CV that is not up to date, or that (b) your LinkedIn profile will be updated over time and that you won't be able to go back and edit your dissertation?
If you mean (a), then no, you need to keep your CV up to date and a link to an online profile will not work. Your academic CV should always be kept as up to date as possible and it should absolutely be updated before you submit it in an application or include it in a dissertation. If you are doing your CV correctly, it will include different information than a LinkedIn profile and there is a strong expectation that every academic will have one.
If you mean (b) and are just worried that an archival copy of your CV will be out of date, sure, add a link to LinkedIn or similar. My CV links prominently to my academic homepage on at a permanent (i.e., non-university) URL, which is kept up to date, and which includes a link to the latest version of my CV at all point. I include a date in the footer of my CV although folks will have a date in your dissertation. Personally, I think this is better than relying on a for-profit company and its URLs for posterity.
In terms of the QR code, I'd skip it. These days, almost everybody who reads the dissertation will read a soft copy. A hyperlink will be much more useful. I suspect that a QR code will just end up make the document look dated at some point in the rather near future.