What might cause "ERROR 000728: Field does not exist within table" in this code?

Confirm that the field FID_Con_st exists immediately before your line of code tries to use it by running this test.

for field in arcpy.ListFields(Intersect_str):
    print field.name
str_Dissolve = arcpy.Dissolve_management(Intersect_str, "in_memory/str_Dissolve", ["FID_Con_st"], "", "MULTI_PART", "DISSOLVE_LINES")

My suspicion, from the fact that you are trying to write your output feature class to an in_memory workspace, is that the Intersect_str variable may be set to something already in memory - and there is no guarantee that a feature class held in memory has the basic set of fields you come to expect in a feature class stored in a file geodatabase.

For background on why I have suggested the in_memory workspace be considered a possible cause see the related bug NIM063795.

The test below shows that an in_memory workspace can be used with Dissolve so I think your question needs to be re-focussed on what you are using as your dissolve field and your code snippet does not explain where that is coming from.

import arcpy

if arcpy.Exists("C:/Temp/Test.gdb"):
    arcpy.Delete_management("C:/Temp/Test.gdb")

if arcpy.Exists("in_memory"):
    arcpy.Delete_management("in_memory")

arcpy.CreateFileGDB_management("C:/Temp","test","CURRENT")
arcpy.CreateFishnet_management("C:/Temp/test.gdb/fishnetPolys","0 0","0 1","1","1","2","2","#","NO_LABELS","#","POLYGON")
arcpy.AddField_management("C:/temp/test.gdb/fishnetPolys","DissolveField","TEXT","#","#","#","#","NULLABLE","NON_REQUIRED","#")
arcpy.CalculateField_management("C:/temp/test.gdb/fishnetPolys","DissolveField","'Any Value To Dissolve On'","PYTHON_9.3","#")
arcpy.CopyFeatures_management("C:/temp/test.gdb/fishnetPolys","in_memory/fishnetPolysInMemory","#","0","0","0")
print "Try dissolving with input and output both on disk ..."
arcpy.Dissolve_management("C:/temp/test.gdb/fishnetPolys","C:/temp/test.gdb/fishnetPolysDissolved","DissolveField","#","MULTI_PART","DISSOLVE_LINES")
print "Worked!"
print "Try dissolving with input in memory and output on disk ..."
arcpy.Dissolve_management("in_memory/fishnetPolysInMemory","C:/temp/test.gdb/fishnetPolysDissolvedFromMemory","DissolveField","#","MULTI_PART","DISSOLVE_LINES")
print "Worked!"
print "Try dissolving with input and output both in memory ..."
arcpy.Dissolve_management("in_memory/fishnetPolysInMemory","in_memory/fishnetPolysDissolvedInMemory","DissolveField","#","MULTI_PART","DISSOLVE_LINES")
print "Worked!"
print "Try dissolving with input on disk and output in memory ..."
arcpy.Dissolve_management("C:/temp/test.gdb/fishnetPolys","in_memory/fishnetPolysDissolvedToMemory","DissolveField","#","MULTI_PART","DISSOLVE_LINES")
print "Worked!"