I would never recommend disabling the alarm, however a malfunctioning alarm is worse than not having one.
There will be a switch in the door near the key cylinder, which is used to disarm the alarm. The switch is usually a simple ground switch that lets the anti-theft computer know that the door was unlocked via the key, and that it's okay for the door to be open and the ignition to turn on. Unlocking the door by pulling the lock button will not activate the switch and therefore the alarm will go off.
Disassemble the door, and you can gain access to this switch, and troubleshoot it with a multimeter. Corrosion or dirt/debris can make this switch not make contact, thereby not allowing you to disable the alarm. Clean the switch, give it a nice coat of dielectric grease, and you should be good to go with the factory alarm providing security.
In the meantime, if you unlock the door from the other side, this should allow you to get past the problem until you have the time to fix it. (Or, open the trunk with the key, unlock and open the door, then close the trunk, which is a pain, but should work.)
Disabling the alarm isn't hard, but pretty much needs the system to be disarmed in order to work.