When I turn off the ignition of my 1999 Honda Accord LX, I have to press "unlock" on my keyless remote in order to be able to start the car. If my battery fails on the remote, I cannot start the car. I do not want to have to activate the unlock button on my keyless remote each time I wish to start the car. What can be done to deactivate this feature?

That sounds to me like an after-market alarm that has failed, or is about to completely fail, and is in need of being fully removed from the vehicle.

Not aftermarket -- factory installed. If you unlock the car doors using the keyless remote you have maybe a minute to start the car without using the "unlock" button on the remote. If you wait longer than a minute after using the keyless remote to get in the car, you have to push the "unlock" button again to start the car...