Be careful here, if you connect the scope ground to 'L' and the Due is powered from USB, you may be shorting 'L' via mains earth (if your local mains supply has this). You're better off 'scoping both the 'L' and 'H' simultaneously.Can (no pun intended) we have 'scope traces of the noise?I assume you've checked that there is in fact 3v3 across the modules supply pins. (I had a board with a 5v part fitted instead of 3v3 and all I got was noise).If you are using the 230 part instead of the 234, note that pin 5 is not supposed to be connected to 3v3, it's a reference output, forcing this to 3v3 may be why it doesn't work as expected (and could damage it).
That particular transceiver has no enable pin so it is essentially always on. If you have signals on CANRX and CANTX but the CANL and CANH lines are garbage then there is likely a wiring fault on the H/L side (the actual CAN bus). What you should get is a square wave on H/L when you use L for ground on your scope and H at the scope input. Or connect to your local board ground with the scope and L will sink then return to center and H will rise and return to center. They ought to rest at essentially the exact same voltage. At rest H = L so you will measure 0 volts. When active they pull away from each other as a squarewave that ought to look like an exact mirror image of each other. The more it differs from that ideal the worse off you are.