Dynamic[Refresh[Import["sieve.txt", "Data"], UpdateInterval -> 1]]

This function will monitor the file modification timestamp to determine if the contents need to be imported again:

Clear[monitorFile]
monitorFile[
  Dynamic[contents_], 
  file_String?FileExistsQ,
  importFunction : _ : Import, 
  formatFun : _ : Identity
] := DynamicModule[{
   modificationDate,
   display = ""
   },
  DynamicWrapper[
   DynamicWrapper[
    Column[{
      Dynamic[modificationDate, TrackedSymbols :> {modificationDate}],
      Dynamic[display, TrackedSymbols :> {display}]
    }, Alignment -> Left]
    ,
    modificationDate;(* This triggers the update *)
    If[ DateObjectQ[modificationDate],
        contents = importFunction[file];
        display = formatFun[contents]
        ,
        contents = display = Missing[]
    ],
    SynchronousUpdating -> False,
    TrackedSymbols :> {modificationDate}
   ]
   ,
   modificationDate = Quiet @ Information[File[file], "LastModificationDate"],
   SynchronousUpdating -> True,
   UpdateInterval -> 1.
 ]
]

Example usage:

file = "table.txt";
Export[file, List /@ Range[5], "Table"];
monitorFile[Dynamic[x], file, Import[#, "Table"] &, Grid]

Note that I'm not 100% sure that Information[File[file], "LastModificationDate"] works on all operating systems since there are a number of different timestamps associated with files depending on the OS. You may have to look at Information[File[file], "Properties"] to see what property you need to query.

Another thing to keep in mind is that Dynamics will only update when they need to be displayed, so you can't use this as a background file monitor. If that's what you need, you should look at ScheduledTask instead.

Update

I made the function more robust in the situation where the file gets deleted suddenly.