Perhaps the following code may help:-
Application.ScreenUpdating = False
Dim lr As Long
lr = Range("A" & Rows.Count).End(xlUp).Row
For Each cell In Range("A1:A" & lr)
If cell <= [Today()] - 7 Then
cell.EntireRow.Copy Sheet2.Range("A" & Rows.Count).End(3)(2)
Application.CutCopyMode = False
Application.ScreenUpdating = True
The code does as you ask and following is a link to my test work book for you to peruse:-
Click on the button to see it work.
I'm not sure if you wanted the data in sheet 1 cleared after the transfer of data to sheet 2 so the code just refreshes the data in sheet 2 meaning that all data in sheet 1 will remain but in sheet 2, the data will show old and new entries. Let us know if you need to change this part.
I hope that this helps.