No because you'll be fetching it from the gateway, and then writing to the same gateway. No cross-domain going on.
But then is it not a security problem that other IPFS apps retrieved through the same gateway will be seen by the browser as being of same origin?
Yes, for now. There is code existing to set the Suborigin header, but browser support is limited, and I'm not even sure if that code is merged into mainline IPFS yet.