You can do this with some cards/tags with rooted Android phone(some tweaks are needed in the NFC subsystem) with certain NFC chipsets. It needs to be card with broken crypto (Mifare) so you can obtain the password and then copy complete card's content. The door reader shouldn't rely on the card's UID (unique serial number) as this can't be spoofed, the chipsets doesn't allow this.

I don't waste time with closed systems, so I can't comment directly on iOS capabilities, but I would guess, that it's going to be almost impossible to make it working in such locked down iOS world.