The burnt resistor provides the common ground between the PLC's RS232 and the other device (usually a PC) RS232. If there is serious ground imbalance then a lot of current can flow through the resistor which caused it to burn. However, this resistor should not affect the PLC normal function. If there is erratic behavior it is likely a failed I/O (digital or analog I/O).
The picture appears to be a T100MD888+ which already EOL in 2011. This unit could be too old to repair and we recommend replacing it with a new model.
The replacement model is FMD88-10 but in order to replace it you will need the original ladder + BASIC software. Otherwise you may need to re-write the program to make it do the function that it is intended.
More information about the FMD88-10 is available at:
http://www.triplc.com/fmd8810.htm