For what its worth (a little late to the game from the original post). I'm replying since I also recently received similar error messages when running "Extract By Mask" in a Python script (ArcDesktop 10.3.1). In my particular instance I am referencing a single DEM and selecting multiple polygons to create unique rasters from each polygon. After realizing that a temporary file is created each time the "Extract By Mask" command is run, executing the save command allowed the passing of a variable with a unique name, eliminating the error messages, allowing the script to create the intended output(s). I also specified a scratch workspace.
script Example:
env.workspace = "//w63-atlas/Stormwater/Waterways/Structure Flood Severity"
bslash = "\\"
Raster = bslash+"\\w63-atlas\Stormwater\AMEC\CLAY\TOPO\DEM\claydem.gdb\claydem"
arcpy.env.scratchWorkspace = "//w63-atlas/Stormwater/Waterways/Rasterdatasets.gdb"
tile = raw_input("Enter FEMA TILE : C4, C5, C6, D4, D5, D6, E5, E6, F5, F6, G5, or G6: ")
prefix = "Export_"
suffix = "_Culv*_1.shp"
depiwant = prefix+tile+suffix
fcList = arcpy.ListFeatureClasses(depiwant)
arcpy.AddMessage(Raster)
for i in fcList:
Culvnum = i[14:-4]
rasnew = gdb+"/"+tile+"rz"+Culvnum[:-2]
outExtractByMask = ExtractByMask(Raster,i)
arcpy.AddMessage(outExtractByMask)
outExtractByMask.save(rasnew)
exit