Why will windshear inop illuminate after landing in an MD-83?
The WINDSHEAR INOP light tells the pilots that the Predictive Windshear System cannot work:


  System Annunciator Lights
  
  [...]
  
  An amber WINDSHEAR INOP annunciator is installed on the annunciator panel.
  This light illuminates any time the windshear systems detects a fault which
  renders the system inoperative.


(MD-80 FCOM Sec. 16 - Instrumentation / Navigation / Communication - Predictive Windshear System)

In order for this system to function, the radar system needs to be active, which is a hazard to the ground crew:


  RDR-4B Radar System Theory
  
  Like all modern radar systems, the RDR-4B operates by emitting short
  intense pulses of microwave energy which are reflected by objects having
  reflective characteristics within the range of the system. The reflected
  signals are processed to produce visual displays that are representative of the
  size, intensity, bearing and distance of the targets. The RDR-4B incorporates
  an advanced microprocessor design which utilizes the Doppler principle to
  identify areas of moderate and higher turbulence as well as low level
  windshear (microburst) activity. The RDR-4B system is designed to operate
  in the windshear mode automatically any time the aircraft is below 1500’
  AGL, at least one engine is running, and the transponder is not in OFF or
  STBY, REGARDLESS OF RADAR MODE SELECTED. The system also
  operates in the windshear mode below 1500’ AGL using an alternate scan
  technique if the radar is operating in any mode (WX / TURB, WX, MAP). The
  windshear mode is activated at 2300’ AGL. However, no alerts or displays
  are annunciated above 1500’ AGL.
  
  WARNING:
  Failure to return the transponder to STBY after landing or
  selecting any mode except STBY prior to leaving the ramp area
  allows the radar to operate in the windshear mode creating a
  radiation hazard to personnel on the ground. The RDR-4B
  radiation hazard area extends 13.4 feet from the radar antenna
  in a 120 degree arc left and right of the aircraft centerline.
  This hazard exists even if the radar mode is selected OFF or
  TEST.


(MD-80 FCOM Sec. 16 - Instrumentation / Navigation / Communication - Predictive Windshear System)

After landing, the pilots will switch the transponder to STBY in order to deactivate the radar system before reaching the apron (where ground crews are working). This will disable the Predictive Windshear System and causes the WINDSHEAR INOP light to illuminate.