Well, in answer to the question briefly, the U.S. did study this in 2003. They sent essentially the same questionnaire to people, with the covering message that it was, on the one hand, voluntary, and in the other case that it was mandatory. The difference in response rates was approximately 20%. In other words, the voluntary survey had about a 20% less response rate. As Statistics Canada does as well, of course they do repeated callbacks. If they wanted to have even vaguely comparable data quality, they needed so many callbacks that it was infinitely more expensive in this case, and so they abandoned it and returned to a mandatory American community survey.
On July 27th, 2010. See this statement in context.