Automatically delete column once the date in that column has expired

I presume the names are in Column A and the dates are in Column B on Sheet1 then, try the following macro:

Private Sub Workbook_Open()
Dim i as long, ilastrow as long
With Thisworkbook.Worksheets("Sheet1") 'replace Sheet1 with the actual sheet name
ilastrow = .range("A" & .rows.count).end(xlup).row 'replace A with the column with names
For i = ilastrow to 1 step -1
if .range("B" & i) > Date then .range("A" & i).EntireRow.Delete 'replace B with the column with dates
Next i     
End With
End Sub

copy this code to the ThisWorkbook Object and enable macros by default for this workbook. Then this code will run everytime the workbook is opened and will delete all expired rows.
HTH!