The following table displays the permissions required by users for Approval Objects:

ObjectallowCreateallowDeleteallowEditallowReadmodifyAllRecordsviewAllRecords
AdhocApprovalGroup__cTRUETRUETRUETRUEFALSEFALSE
AdhocApprovalProcess__cTRUETRUETRUETRUEFALSEFALSE
AdhocApprover__cTRUETRUETRUETRUEFALSEFALSE
ApprovalJob__cTRUETRUETRUETRUEFALSEFALSE
ApprovalProcessInstance__cTRUETRUETRUETRUEFALSEFALSE
ApprovalRuleAssignee__cFALSEFALSEFALSETRUEFALSEFALSE
ApprovalRuleDimension__cFALSEFALSEFALSETRUEFALSEFALSE
ApprovalRuleEntry__cFALSEFALSEFALSETRUEFALSEFALSE
ApprovalRule__cFALSEFALSEFALSETRUEFALSEFALSE
Approval_Matrix__cFALSEFALSEFALSETRUEFALSEFALSE
Approval_Process__cFALSEFALSEFALSETRUEFALSEFALSE
Approval_Request_History__cTRUETRUETRUETRUEFALSEFALSE
Approval_Request__cTRUETRUETRUETRUEFALSEFALSE
Apttus__APTS_Admin__cFALSEFALSEFALSETRUEFALSEFALSE
Apttus__APTS_Agreement__cTRUETRUETRUETRUEFALSEFALSE
Apttus__Term_Exception__cFALSEFALSEFALSETRUEFALSEFALSE
AsyncActionInfo__cTRUETRUETRUETRUEFALSEFALSE
Backup_Approver__cTRUETRUETRUETRUEFALSEFALSE
FormulaField__cFALSEFALSEFALSETRUEFALSEFALSE
Global_Discount_Policy__cFALSEFALSEFALSETRUEFALSEFALSE
OpportunityTRUETRUETRUETRUEFALSEFALSE
ReminderInstance__cTRUETRUETRUETRUEFALSEFALSE
Reminder__cFALSEFALSEFALSETRUEFALSEFALSE
SearchFilter__cFALSEFALSEFALSETRUEFALSEFALSE
Term_Exception_Approval_Condition__cTRUETRUETRUETRUEFALSEFALSE
Term_Exception_Approval__cTRUETRUETRUETRUEFALSEFALSE

For information on setting object permissions, refer to this Salesforce article.