I just bought a scarf today and thought the care label was way too big so carefully snipped it off, just leaving the more discreet Burberry label. I will keep it and if need be I can stitch it on again. Though really, I can't ever see the need to re-attach it. I wont be on-selling it as I can tell I will wear it for a very long time :)
I have the crinkle scarf so I used very fine thread and sewed the hanging part onto the scarf. So it's there, but not flapping around when I wear it.
I got the children's scarf for my cousin, and I looped the tag and sewed the end on itself close to where it's sewn in on the scarf. That way she knows its dry clean only, and that it's out of the way.
It doesn't take that much thread and it's barely noticeable if done right.
I usually tear off the large tag because it just looks tacky, especially on the silk crinkly scarfs. Some times I leave the smaller one that says Burberry, but I don't necessarily need it to be there. Along as I know my scarf is authentic, that's all that matters. It seems that based on the way the tags are sown on that they are meant to be taken off anyways.
I keep the tag on. On my cashmere Burberry scarves, it isn't obvious, but on my crinkle one, it always shows so I safety pin it up so it's flat and doesn't hang down. Safety pinning it has worked so far. I just like to keep the tag on for authenticity in case I ever decide to sell it.