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Generally, no. Canada has no general sales tax refund for visitors and provinces that administer their taxes, only allow this is very specific and limited cases. It certain provinces, the saves tax is taken as part of the HST (Harmonized Sales Tax) which is administered federally, so any rebate would have to be allowed at the federal level.
For specific rebates, some provinces do provide some exemption. For example, Quebec has a rebate for sales tax on movable property (road vehicles, mobile homes, etc). Here are the official rules.
What is more common is when buying goods online, a store which does not have a presence in the province where products are shipped, does not charge that provinces local tax, even if the province where the product is sold from is using harmonized sales tax.
As you can see, for typical travel between provinces, none of this applies, so the answer to your title question is mostly no.