I hope your FilteredEqDump table has an ID column...
Then use a sub-query to get the IDs and use those IDs in the WHERE clause:
Delete from FilteredEqDump
Where FilteredEqDump.ID IN
(SELECT FilteredEqDump.ID
FROM FilteredEqDump
inner join AssociateProductionDetail on
FilteredEqDump.EmployeeCode=AssociateProductionDetail.EmployeeCode and
FilteredEqDump.CountyTaskFlag=AssociateProductionDetail.CountyTaskFlag and
FilteredEqDump.ProjectName=AssociateProductionDetail.ProjectName
inner join NonProdWorkType on
FilteredEqDump.EmployeeCode=NonProdWorkType.EmployeeCode and
FilteredEqDump.Employeename=NonProdWorkType.EmployeeName and
FilteredEqDump.CountytaskFlag=NonProdWorkType.CountyTaskFlag and
FilteredEqDump.CountyGroupname=NonProdWorkType.CountyGroupName and
FilteredEqDump.ProjectName=NonProdWorkType.ProjectName
where FilteredEqDump.WorkType ='NON PRODUCTION')