The image doesn't have to be saved, but is it still possible to resize it? Would I have to cache it?
It can be done with CSS. This blog explains how: http://blog.sachinkraj.com/how-to-resize-images-with-css/
It would have to be loaded into memory at very minimum, but there's no reason you couldn't load it into memory from a remote location (file_get_contents or cURL or whatever), manipulate it and spit it back out to the client.
As far as performance goes, unless the image can be accessed over a very high speed connection (LAN level fast), it's probably going to cause a very significant lag in processing the image. To alleviate this hit, you could cache it, but no, you would not have to. (And if you were going to cache it, it would likely be better to cache the processed version.)