Well, you don't want your IFs to be nested like that with all the closing brackets at the end. Try this
List(", ",
If([Engr check]=true and [Engr Approval Status]<>"Approved",UserToName([Engineering Manager Assigned])),
If([Production Check]=true and [Production Approval Status]<>"Approved",UserToName([Production Manager Assigned])),
If([Quality Check]=true and [Quality Approval Status]<>"Approved",UserToName([Quality Manager Assigned])),
If([Additional Check]=true and [Additonal Approval Status]<>"Approved",UserToName([Additional Approval Assigned]))