2007 Saturn Aura; when trying remote start horn beeps three times but doesn't start. I've replaced battery. Other functions work fine - door locks, trunk opens, alarm arms. Other key fob works fine.

Press the lock button on the remote then press the remote start button. if this does not work check remote start status using steering wheel buttons

To expand on rickb63, the tones are indicating that the remote start cannot be activated. The usual culprit in this situation is the hood latch. There is an electric switch that tells the car that the hood is closed. If the hood is opened during the remote start sequence, the vehicle will stop. Likewise, a malfunctioning switch will prevent the remote start from engaging.