"Should house prices be included in the CPI?" is not a good question to ask.
The best answer to that question is another question:
"Why do you want to know?" Or, "It depends; what are you planning to use the CPI for?"
Instead, it would be better to ask the question in a different way -- one that already answers that second question. Like:
"Should house prices be included in the price index that the Bank of Canada uses to define 'inflation' when it targets inflation?" (They aren't really included now, except very indirectly over the longer term -- long story (pdf).)
It is easy to sketch a model where house prices should be included in the price index the Bank of Canada targets, or where only house prices should be included in that index.
Suppose that nominal house prices were sticky, but that all other prices (including wages) were perfectly flexible. Then you could make a good case that the Bank of Canada should target the inflation rate of house prices only. The argument follows directly from the idea of "targeting the stickiest price".
Suppose there were a real shock that would require the relative price of houses to rise to restore equilibrium.
If the Bank of Canada targeted house price inflation, the nominal prices of all other goods would fall as a result of that real shock. But since, by assumption, the nominal prices of all other goods are perfectly flexible, they would fall instantly, restoring equilibrium instantly. A fall in the price of all other goods raises the relative price of houses, without the nominal price of houses needing to do anything. Which is good, because the nominal price of houses can't do anything quickly, by assumption.
If instead the Bank of Canada targeted inflation in all the other prices, and ignored house price inflation, the same real shock would cause an excess demand for houses, because the actual relative price of houses would rise only slowly towards the new equilibrium price, and so would be below that equilibrium price in the meantime. There would be a housing shortage.
If we make the exact opposite assumption, that house prices are perfectly flexible and all other prices are sticky, we get the exact opposite policy recommendation. The Bank of Canada should target all other prices, and ignore house price inflation. Because if the Bank of Canada targeted house price inflation (or even just included house prices in the index it targeted), a real shock that required the relative price of houses to rise would cause a recession. Because it would require the prices of all other goods to fall, and they can't fall instantly, by assumption, so those prices would be higher than equilibrium in the short run, which means all other goods would be in excess supply. (The market for houses would always be in equilibrium, regardless of what the Bank of Canada targeted, since house prices are perfectly flexible.)
What about the empirical evidence?
Eyeballing the Teranet-National Bank composite 11, (click on "Historical Charts - National Composite Index Values and Sales Pair Count" on the right side of the page) we see that: house prices rose more quickly than normal from early 2006 to mid 2008; then reversed course and fell quickly to early 2009; then reversed course again and rose more quickly than normal, with a gradually slowing rate of increase.
This gives us a rough idea of how the Bank of Canada's policy would have been different than it actually was, if house prices had been included in the price index it used for inflation targeting. Monetary policy would have been: a little tighter than it actually was from early 2006 to mid 2008 (which would have been good); then looser than it actually was until early 2009 (which would have been very good); then a little tighter than it was until recently (which would have been bad).
Overall, this suggests that monetary policy would have been better than it actually was if the Bank of Canada had been targeting 2% inflation for a price index that included house prices properly. But not unambiguously better, because monetary policy would have been tighter during the slow recovery. And how big those differences would have been would depend on how big a weight house prices had in the price index.
The house price guard dog, in other words, barked correctly two times out of three. The NGDP guard dog did better, barking correctly all three times. And the NGDP dog barked loudly enough for us to hear, if we had been listening. Unless house prices had a big weight in the price index, we might not have heard the house price dog over all the other price dogs.