See here, where the argument is made that Canada's museums are being starved of needed monies. Let us suppose that is actually true. Is there an obligation on countries to preserve their heritage through museums? There's one argument I'm clearly not going to buy on this score (see here), in that whatever is valuable in the past need not be preserved in its own right; so, there must be some instrumental value in preserving the past. The obvious candidate is educational: historians and citizens would benefit from having the knowledge of what has gone before in their countries. That seems to me to be right, but pretty thin justification for public museums -- it's a justification for museums generally.
I would suggest, though, that since education is expressly a provincial function in Canada, then the provinces should be responsible for coughing up the money to pay for museums (perhaps part of the federal-provincial transfers?). So, the moral argument only gets us to "someone" should keep museums open; the legal argument gets us all the way to public museums; but there doesn't seem to be a way to get us (yet) to the feds having any responsibility.
However, on the basis of expediency, it seems that final step can be made: the federal government controls vastly more money than the provinces, and is uniquely positioned to see what would be of benefit to the history of the country overall, rather than the potentially more parochial considerations of the provinces. That, then, would be the argument for federal funding of public museums (as well as an argument that the provinces need to be doing more).