Just had this problem on my 99 about 3 months ago. Had half the interior out trying to trace it down.
First, try to unlock the doors using the rocker switch in the back. If the doors unlock or you hear the relay click, the relay is okay. If they don't unlock, check the relays for the PAT system located in front/above the right rear wheel well to be sure they are working. Since your remote doesn't unlock the doors either, good chance the relay may be bad.
If the relay works, then you probably have a broken wire connection. There's different colored wires for each function of the unlock relay so you'll have to determine which color does what. If you find the relay is good and can't determine the wire colors to look for, post again and I'll look them up. Think I wrote all this stuff down when tracing my problem if I can find it. Colors are probably the same for my 99 as they are for your 01.
Once you determine the wire color to look for, there is a connection for all the unlock wires located in the channel under the bottom door panel plastic trim of the right rear passenger door. Pry the bottom panel off and look through the wire bundle toward the rear. Look closely for the connection as its hard to see among all the other wiring. There should be 3-4 wires of the color you're looking for connected somewhere. There's an insulation wrap over the connection that you will have to remove to check the splice. Mine had corroded and the splice broke.
