The reason your macro is slow is because however good VBA is it is very very slow at interacting with the worksheet. I just did a quick test on my computer and the time taken to interact with one cell was about 0.1 milliseconds. You are interacting with 15000 cells individually about 3 or four times in your loop so that is going to take a least a minute.
What is interesting is that you can load all the contents of a worksheet into memory in almost the same time as it takes to load a single cell. So the way to speed up you macro by a factor of 500 or so is to load the source worksheet into a variant array and then do the copying between this array and an output array and then write the whole array back, just two accesses to the worksheet so it will be done in well under a second. So try this code:
Note I have got rid of the things which you put in it to try a speed it up which are not necessary with this code:
Sub CopyYes()
Dim c As Range
Dim j As Integer
Dim Source As Variant
Dim Target As Variant
' Change worksheet designations as needed
Source = ActiveWorkbook.Worksheets("data").Range("A1:BH15000")
Target = ActiveWorkbook.Worksheets("Issue Comparison").Range("A1:BH15000")
j = 4 ' Start copying to row 1 in target sheet
For i = 4 To 12000 ' Do 12000 rows
If Source(i, 60) = "Review" Then
For k = 1 To 60
Target(j, k) = Source(i, k)
Next k
j = j + 1
End If
Next i
ActiveWorkbook.Worksheets("Issue Comparison").Range("A1:BH15000") = Target
End Sub