You can use either polygons or points, which you choose depends on whether you need to update large areas or just individual cells.
Once you've got your points/polygons created with an attribute table column with the values you want to use to replace values in the DEM with, convert the feature class to raster with the Feature to Raster tool (or the Point to Raster/Polygon to Raster tools). Make sure you set the appropriate environments - Processing Extent->Extent and Snap Raster and Raster Analysis->Cellsize - to your DEM and set the correct column as the value field in the tool.
You can then use a statement like the following in the Raster Calculator:
Con(IsNull("newvalues"), "DEM", "newvalues")
This means: If a cell in the "newvalues" raster is NoData, use the "DEM" cell value, else use "newvalues" cell value.