To conduct an OR within a Sumproduct utilise the + operator (if you have numerous test you can switch to an ISNUMBER(MATCH test with an inline array of accepted values))
=SUMPRODUCT(--(Timesheets!$E$2:$E$1000=$B3),(Timesheets!$D$2:$D$1000="001")+(Timesheets!$D$2:$D$1000="006"),Timesheets!$V$2:$V$1000)
Given 001 + 006 can not exist in D simultaneously the above should work without incident.
It would still be quicker to conduct 2 SUMIFs using Concatenation (IMO)
=SUMIF(Timesheets!$Y$2:$Y$1000,"001:"&$B3,Timesheets!$V$2:$V$1000)+SUMIF(Timesheets!$Y$2:$Y$1000,"006:"&$B3,Timesheets!$V$2:$V$1000)
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