I think, the only reason you could get a warning sound the key is in the ignition is if a door is open. (close all doors)

Does the 03 civic have a main relay like my 91 crx si?
(check the wiring diagram)

Does the check engine light come on when you switch the ignition to ON (don't go all the way to START)?
three possibilities:
-no check engine light at all (power to ECU or bad ECU)
-check engine light comes on an stays on (get an ECU code reader and find out what the ECU thinks is wrong)
-check engine light comes on then goes out after a short time (ECU is functioning correctly)