No, that is not always the case. If you run OBDII fuel injection, it will trip a code in your computer if you don't have cats. But will still run fine. I rodded out the cat in my 2002 Grand Am GT when it became clogged (no smog check here), and although it lit the check engine light, it hasn't hurt driveability one bit.
If you're running an EFI that requires OBD1 instead, then you're fine without catalytic converters. An example of that would be the L98 tuned port fuel injection from a Corvette.
