Try to write the NetworkPath as a variable in the formula:
.Range("C2").Formula = "=SUM(" & Network_Path & "\[Source_File.xlsx]Data'!$B$2:$B$20)"
Concerning the general debugging of formula with variables, try the following simple trick:
Public Sub TestMe()
Dim np As String 'do not use "_" in file name FWIW
np = "C:\Users\" 'or something else
Debug.Print "=IFERROR(INDEX('" & np & "[Source_File.xlsx]Data'!A:A; MATCH('Sheet1'!C3;'" & np & "[Source_File.xlsx]Data'!R:R; 0)); "")"
End Sub
Then see what is present in the immediate window. Press Ctrl+G to open it. The result from the current formula is:
=IFERROR(INDEX('C:\Users\[Source_File.xlsx]Data'!A:A; MATCH('Sheet1'!C3;'C:\Users\[Source_File.xlsx]Data'!R:R; 0)); ")
Thus, you probably have to find a way to remove the last " sign:

Then try to copy the formula to your Excel file and if it works, then it is ok.
Another option is to try to do the exact opposite. E.g., write the formula in Excel, as it should be written and make sure it works. Then select the cell with the formula and run the following:
Public Sub PrintMeUsefulFormula()
Dim myFormula As String
Dim myParenth As String
myParenth = """"
myFormula = Selection.Formula
myFormula = Replace(myFormula, """", """""")
myFormula = myParenth & myFormula & myParenth
Debug.Print myFormula
End Sub
It will print in the immediate window the formula, as it should be used in VBA. Then replace the path with a variable and give it a try.