Is it possible to burn CD/DVD with iPhone and USB CD/DVD Writer?
It's unlikely you can burn a DVD since I doubt that is a process an iPhone would support. What might work though is DVD-RAM.
DVD-RAM is not as popular as DVD-RW and DVD+RW, likely because the media costs more. Or maybe it costs more because it's not as popular. Either way there is a big difference in DVD-RAM in that it does not have a spiral cut track like other formats, it has sectors and blocks in concentric tracks like a hard drive. Because of this it acts very much like a hard drive to a computer, as in files aren't "burned" they are just copied over to the file system. Because DVD-RAM shares so many features with other DVD formats most devices with a DVD drive can read them.
I have two "newish" USB optical drives, one of which supports DVD-RAM, so shop carefully if you want to experiment with this. One drive is the Apple USB Superdrive that Apple has been selling as an accessory for quite some time now. Plugging it into my MacBook and looking at it's supported media in System Information I see that DVD-RAM is not supported. With my other USB DVD drive, bought from Other World Computing years ago, I see that this drive does support DVD-RAM. The drive in the case is listed in System Information as model HL-DT-ST DVDRAM GH24NSB0 if that helps.
I can't find my DVD-RAM discs right now to experiment with this myself. The comments on looking out for the power draw is something I have concerns over also. You will want the powered Lightning to USB 3 adapter and a drive that does not rely on USB for power. Even though the drive won't take power from the USB port the USB adapter in the drive might, and it could be more than an iPhone can provide without being plugged in to a power brick.
While I have not tried DVD-RAM discs with my iPhone yet I will mention this as something that I believe has high chances of allowing file transfer into and out of an iPhone. If I can find all the parts I need soon then I'll come back to report on if my experiment was a success.
As far as I know there is no CD equivalent to the DVD-RAM. There are combinations of CD drives and software that can mimic this behavior but that's outside of the standardized specifications for the compact disc, and therefore unlikely something an iPhone would support.