I'm not sure why that isn't working for you, but why not try an additive approach instead of a removal approach (see code below). I confess I haven't tested this code to make sure it will work, but I'm pretty sure it will.
EDIT: I've tested this script out on a test dataset and it works fine. I'll continue investigating what the problem is with your original script.
EDIT 2: I've tested out a simplified variant of your original script step-by-step to see where it was failing, and I couldn't find anything wrong with the procedure. My one thought is that it appears that your
fieldmappings.removeFieldMap() statement isn't properly indented. However, if that's the case in the script you are running, it should simply throw an error.
fieldList = ["ROUTE_ID","TRIP_NUM","OFFICE_ID","VOLUME","ACTIVITY","ARRIV_TIME"]
listEventTables = arcpy.ListTables("RS_EVENT_ALL_*")
fieldMappings = arcpy.FieldMappings()
#iterate through fields and create unique fieldmap for each
for field in fieldList:
#creates a fieldmap with name of input field
vars()[field] = arcpy.FieldMap()
for table in listEventTables:
outField = vars()[field].outputField
outField.name = field
vars()[field].outputField = outField
eventTable = outGDB + "Merge" # Merge of event tables (has to be in a GDB)
arcpy.Merge_management(listEventTables, eventTable, fieldMappings)