Are there any circumstances under which the SD card can be removed from the device while powered up?
Perhaps it could be possible to operate only from RAM temporarily? This could allow a different SD card to be plugged in to copy some data to RAM or to the web before plugging the original card back in.
If it's currently not possible, could it be done with some careful software changes?
Well, it's not safe if any filesystem present on SD card is mounted (especially if it's the root filesystem that is mounted). This is true by default in most distributions currently available for Raspberry Pi. But it is not strictly needed. You can run your system from RAM (either by copying filesystem image to tmpfs mounted filesystem or by using initramfs) or some external harddrive or even from network. If you do so, your SD card can be swapped without problems (just ensure to unmount all filesystems mounted on it).
The only problem is that Raspberry Pi has a small amount of RAM, so if you plan on running it this way, you should have a really small image (compressed filesystems like SquashFS may be handy here), memory split that gives ARM most RAM and probably 512 MB version of Raspberry Pi. But that's definitely possible.
As far as I know, it's not safe to remove the disk the operating system is running on (SD card, hard disk drive, etc.) while live.
You may like to look into UnionFS, combined with a RAM disk. Linux live CDs run predominately out of RAM, but I'm not sure how much might be needed in your case.
While SD cards are electrically safe to remove from the Raspberry Pi, the bootloader and root filesystem are currently mounted there, so without some major changes, it isn't safe to remove the SD card while powered up.
If you need to read another SD card, you can attach a USB-based card reader to read from other cards, as drivers for external storage should be present.
