Move a row from one worksheet to another when a date is entered in the Date Claimed column

Well, the first thing you should know about VB is that is pretty spreadsheet specific, so we'll be able to little without an attachment provided.
I'd recommend scrubbing your data and providing at least a rough outline.

That said, you'll need to utilize a worksheet change event to continuously run through sheet1 every time something changes. When it does, it'll copy the relevant data. We toss in a filter to re-sort page 1 to fill in the gaps after the cut & paste.

This code goes directly on the worksheet instead of in a module.

Private Sub Worksheet_Change(ByVal Target As Range)

If Target.Column = 9 And Target.Row > 1 Then
    Target.EntireRow.Cut Sheets("Found").Range("A" & Rows.Count).End(xlUp).Offset(1)
        With Worksheets("Lost").Sort
            .SetRange Range("A2:I99")
            .Orientation = xlTopToBottom
            .Apply
        End With
End If

End Sub

Sure. We just make a few small changes. We move the targeting to > 5, change the method to copy and paste to preserve formatting, and adjust the range of the re-sorting.

Private Sub Worksheet_Change(ByVal Target As Range)

If Target.Column = 9 And Target.Row > 5 Then
    Target.EntireRow.Copy Sheets("Resolved").Range("A" & Rows.Count).End(xlUp).Offset(1)
    Target.EntireRow.ClearContents
        With Worksheets("Unresolved").Sort
            .SetRange Range("A6:I1900")
            .Orientation = xlTopToBottom
            .Apply
        End With
End If

End Sub

It's because you're typing on random lines with no data. It's copying each record directly below the last entry in Column A of Resolved.

If there's no data in A on the resolved then it's put the next record directly on top of it. Put something in A of unresolved first, then add a date in I.

Glad I could help out.