It's rather complicated because of the way cellular communications devices are distributed and how Apple does serious warranty repairs as replacements. The warranty terms for iPhones always notes that warranty service may be restricted to the country of original purchase. However, batteries and screen assemblies are universal parts regardless of country, so it might be possible to get one repaired under warranty in another country.
However, battery replacement isn't covered as a warranty item unless the battery is "defective". If something happens such as the battery failing or severe loss of capacity (80% is the standard) during the warranty period, it may be replaced under warranty, even if it's in another country. If the battery swells, then that would likely damage the device and that's a bit different because it's not a simple part to be replaced.
However, Canada is close enough to the US that it's possible to get it repaired in the US.