You would need a dedicated computer to be used as an FX processor. And depending on the plugins used your mileage may vary. The key here is to not have any harddrive activity, like playback tracks, on the machine and only have it doing DSP. I think you will find that you will get around 256 samples of latency for reliable performance with heavy usage.
If using a RME HDSP Series product, like the Fireface 800, this is indeed a possibility...
THE HDSP series comes with a mixer (also known as TotalMix) which offers zero latency monitoring. This mixer lives "outsidde" of your recording application...very much like having an external mixer sitting on your desk. TotalMix allows the re-routing of all your inputs and playback channels so that you can create FX Sends and headphone monitoring submixes on the fly and recall them at a moments notice with 8 presets per session. There is 0% CPU hit on the computer using this mixer as it is all done by the DSP on the unit. There is also Dimming, Mono Summing, Mackie Control Support, M/S processing, Mute/Solo, Pan, Volume, phase inversion(currently only FF800) and full metering.
Also, using any soundcard will work at zero latency when using an external mixer. The key here is to not monitor your live input through the DAW's software mixer. You must only monitor through your external mixer. All DAW's will automatically create an offset to record your input signal in time with the existing material. You can even use plugins on your playback channels. It is only LIVE INPUT that latency ever becomes an issue...and that includes VSTi's as well, hence the need for low latency monitoring drivers such as ASIO. Remember that latency is only an issue if you want to monitor your FX or Instruments in your DAW on live input. If you are using external FX this is not a problem if you are using an external mixer or something like RME's TotalMix.
So...Yes, you can use your computer like a tape deck. If it wasn't possible then how did people do this stuff 10 years ago before ASIO existed???