These devices are very sensitive to noise on the power supply, as well as EMI. I recently used a similar device and ended up with 100 ohm resistor with a 47uF cap as a LPF for the Vsupply. I believe the application circuit in the datasheet also has this filter (but with no part values). A good start would be a 47ohm resistor with a 10uF cap.
With that said, it will still probably get false triggers on the output, and your software should be able to cope with these. When a remote is sending a signal the AGC kicks in and reduces the gain, so you won't get false spikes in the middle of a transmission, only when its idle. Your software should be able to distinguish between a valid signal (NEC, sony, RC-5) and false triggers.