Are there types of passive RFID tag which can connect to control logic?
It's possible, but most (if not all) commercially available tags just transmit their code upon reception of the carrier. The antenna is their only I/O.
What you need is a low-power microcontroller which is fed from the energy it receives from the reader's carrier. As a microcontroller it can do anything you like, including sending a code to the reader, and switching any external devices like LEDs. Of course these devices will need their own power source, since the power extracted from the carrier will be hardly enough to power the microcontroller. A good low-power microcontroller is TI's MSP430, "ultra low-power" in TI's marketing speak.
You'll need to have good antenna coupling to get a high enough voltage not only to operate the microcontroller, but especially to control the external logic. You may have to use level shifters which accept very low input voltages. For instance the Fairchild FXLP34 can work with input voltages down to 1V.  

This gal made an RDIF tag with an AVR microcontroller and some clever tricks.