This one updates the degrees:

DynamicModule[{inputList, inputData = 0, plotData = {}, 
  dataSet3 = {0, 0, 0}, dataSet1, dataSet2, dataSet4},
 inputList = {{1, 5}, {2, 4}, {3, 8}};
 Dynamic@Column[{InputField[Dynamic[inputData]],
    Dynamic@Button["Plot",
      dataSet1 = Map[Norm, Differences[inputList]];
      dataSet2 = Exp[-N[Pi]*0.345*5*dataSet1/(10*2000)];
      dataSet3 = Quantity[FoldList[10 + (#1 - 10)*#2 &,
         inputData, dataSet2], "DegreesCelsius"];
      dataSet4 = Transpose[{inputList[[All, 1]], dataSet3}];
      AppendTo[plotData, dataSet4]],
    Spacer[5],
    PopupMenu[
     Dynamic[popupSelection], {"DegreesCelsius" -> "C", 
      "DegreesFahrenheit" -> "F"}], 
    ListPlot[
     plotData /. dgs_Quantity :> UnitConvert[dgs, popupSelection], 
     Joined -> True, ImageSize -> {300}, Frame -> True]}],
 SynchronousInitialization -> False]

The UnitConvert is only done at the beginning. It is not dynamic.