We did the Alaska Highway twice, once in 1992 for the fifty year anniversary and again in 2008. We didn't do the Dempster but in 08 we did the top of the world highway and went to the end of the road in Alaska to the west.
I did lose a tire in 08 and found a replacement in Alaska. My recommendation would be if you have tires that aren't mainstream (for example 19.5's) you may want to take an unmounted replacement in addition to your spare and be aware that all tire shops may not be able to mount and balance your particular size. I've had this problem even in the US.