Brooke:
As to the possibility of Her Majesty, whom God save, depriving her son of his dukedom, I am inclined to say no, but, of course, that entirely depends upon the extent of the Queen's displeasure. As it is, being deprived of use of "His Royal Highness" is a huge slap in the face, and a shame that the Duke of York will bear for the rest of his days. Among the most recent precedents for a Prince being deposed was Ernest Augustus Christian of Hanover, Earl of Armagh, and son of Ernest Augustus Wilhelm, 3d Duke of Cumberland and Teviotdale, who was deposed as a Prince of the United Kingdom on 28 March 1919 by the Titles Deprivation Act 1917, as Malcolm said above. Although the titles were deprived, the Dukedoms of Cumberland and Teviotdale and Albany have surviving male claimants who could petition for their restoration if they so chose.
That said, the style of Royal Highness is different from the title of Prince, and it is my understanding that only his style is affected, which leads me to wonder if the original letters patent would be altered. It is a shame that this will cast a cloud over Her Majesty's 70th regnal year, which starts on 6 February.
The last hereditary titles to be deprived were the Dukedoms of Cumberland and Teviotdale (of which Ernst Augustus Prinz of Hanover is 6th de jure Duke), and Albany (of which Carl-Eduard Prinz von Sachsen-Coburg und Gotha is 6th de jure Duke), and the Viscountcy of Taaffe (which fell extinct upon the death of Richard, 13th de jure Viscount, in 1967).
As it is, I can't speculate if this will go further (it's not often that one sees the Queen displeased), but the current Duke of York's reputation is forever damaged. I wouldn't be surprised if the dukedom, to which Prince Louis would one day be entitled, skipped a generation, depending on how damaging the fallout from this is.