One thing you can do is speed up your code, there's a few things I'd change about it in its current state,
It's really slow to access range objects and their value, you should instead load the ranges into a variant array and cycle through the arrays
If you find a duplicate, you still go through and check every other range in both arrays which wastes time, you should skip to the next range once you've found a duplicate
With that in mind I've rewritten your code like this, it's completely equivalent and runs in less than a second on my machine:
Sub dup()
Dim i As Integer, j As Integer
Dim RejectVals As Variant
Dim ScreenVals As Variant
Dim FullDataVals As Variant
Dim SheetName As Variant
Dim output() As String
'Push column on 'Screener' sheet to the right to make space for new output
Worksheets("Screener").Range("A7:A15").Interior.ColorIndex = xlNone
Worksheets("Screener").Columns("B:B").Insert Shift:=xlToRight, CopyOrigin:=xlFormatFromLeftOrAbove
'Pull the values from your 3 ranges into arrays to avoid costly cycling through ranges
ScreenVals = Application.Transpose(Sheets("Screener").Range("A7:A2000").Value)
RejectVals = Application.Transpose(Sheets("Rejected").Range("A7:A2000").Value)
FullDataVals = Application.Transpose(Sheets("Full Data").Range("A7:A2000").Value)
'Resize output column to be same size as column we're screening because
'we're going to place it in the column adjacent
ReDim output(LBound(ScreenVals) To UBound(ScreenVals))
'Cycle through each value in the array we're screening
For i = LBound(ScreenVals) To UBound(ScreenVals)
'Skip without checking if the cell is blank
If ScreenVals(i) = vbNullString Then GoTo rejected
'Cycle through each value in the 'FullData' array
For j = LBound(FullDataVals) To UBound(FullDataVals)
'If it's a duplicate then
If ScreenVals(i) = FullDataVals(j) Then
'Set the relevant value in the output array to 'Reported'
output(i) = "Reported"
'Colour the cell on the 'screener' page
Worksheets("Screener").Cells(i + 6, 1).Interior.ColorIndex = 5.5
'Skip checking more values
GoTo rejected
End If
Next j
'Next cycle through all the 'Rejected' values
For j = LBound(RejectVals) To UBound(RejectVals)
'If it's a duplicate then
If ScreenVals(i) = RejectVals(j) Then
'Set the relevant value in the output array to 'Rejected'
output(i) = "Rejected"
'Colour the cell
Worksheets("Screener").Cells(i + 6, 1).Interior.ColorIndex = 4
'Skip checking any more values
GoTo rejected
End If
Next j
rejected:
Next i
'Pop the output array in the column next to the screened range
Worksheets("Screener").Range("B7:B2000") = Application.Transpose(output)
End Sub
I check for duplicates in the 'Full Data' sheet first which means if there is a duplicate in both tables then it will default to 'Reported' and a yellow cell, if you'd like the opposite you can swap the order of the loops.
Let me know if there's anything you don't understand