Remove a specific element of a TimeSeries?

You can use TimeSeriesResample as follows:

ts2 = TimeSeries[{0.0, 0.0, 0.0, 0.0, 15.0, 0.0, 0.0, 0.0}, {1}];
pks = FindPeaks[ts2];
ts3 = TimeSeriesResample[ts2, { Complement[ts2["Times"], pks["Times"]]}]

Normal@ts3

{{1, 0.}, {2, 0.}, {3, 0.}, {4, 0.}, {6, 0.}, {7, 0.}, {8, 0.}}

Complete automation

I think what you are actually looking for is DeleteAnomalies. Define these data:

ts=TimeSeries[{1,2,3,15,3,2,1},{1}]

Now compare old and new:

DateListPlot[
    {ts,TimeSeries[DeleteAnomalies[ts["Path"]]]},
PlotMarkers->Automatic,PlotRange->All]

Surgery

Now if you want to play surgeon, you can define a function:

deleteTS[ts_,pts_]:=
DeleteMissing[TimeSeriesInsert[ts,TimeSeries[Thread[{pts,Missing[]}]]]]

It uses TimeSeriesInsert to replace elements at positions pts with Missing[] and then removes them with DeleteMissing. For example compare:

ts["Path"]
deleteTS[ts, {4}]["Path"]
deleteTS[ts, FindPeaks[ts]["Times"]]["Path"]

Out[1]= {{1, 1}, {2, 2}, {3, 3}, {4, 15}, {5, 3}, {6, 2}, {7, 1}}
Out[2]= {{1, 1}, {2, 2}, {3, 3}, {5, 3}, {6, 2}, {7, 1}}
Out[3]= {{1, 1}, {2, 2}, {3, 3}, {5, 3}, {6, 2}, {7, 1}}

You can also delete a set of points:

DateListPlot[
    {ts,deleteTS[ts,{3,5}]},
PlotMarkers->Automatic,PlotRange->All]

and add some fancy replacement/smothing for deleted points:

DateListPlot[
    {ts,
    TimeSeriesResample[deleteTS[ts,{4}],
    ResamplingMethod->{"Interpolation",InterpolationOrder->2}]},
PlotMarkers->Automatic,PlotRange->All]