I've not seen that warning before. I just tested on a file I downloaded, which had the 'Zone.Identifier' stream. Using Explorer, I copied it to a FAT32 volume, then back to my NTFS drive. Sure enough, it lost the 'Zone.Identifier' stream, and there was no warning when I opened it.
This is on a fairly normal Windows 10 installation. YMMV on different versions, of course.