Modbus communication between Arduino and Unitronics Jazz2 PLC
Does the PLC and Arduino have the same ground level? 

What does the signals look like on the oscilloscope when PLC and Arduino try to communicate?

In other words - are you sure that the DTR/DTS lines are the cause for the problem? Try to replicate the scenario that is seen when you connect the units to the PC, probe the communication lines, and check if they look the same.