FEAR

This is a red flag.

Do Until IsSorted(collectionObject)

    ' Sorting algorithm 

Loop

Your algorithm should return a sorted sequence and should not need to be checked. If it fails the check you should rewrite your sorting algorithm and not simply try to do it again.

Abstracting Methods

This bit of code is used often enough to merit it's own sub routine

Sub AssignUnknown(ByRef dest As Variant, ByRef src As Variant)

    If IsObject(src) Then
        Set dest = src

    Else
        dest = src

    End If

End Sub

Now your code looks so much better! It looks like a bubble sort but

For i = 1 To collectionObject.count
    index = i

    AssignUnknown item, collectionObject(i)

    For j = i To collectionObject.count

        AssignUnknown innerItem, collectionObject(j)

        If item > innerItem Then

            collectionObject.Add item, After:=j
            collectionObject.Remove index
            index = j

        End If

    Next j

Next i

Algorithm

it doesn't work, hence you needing to loop until it is sorted. You are only a few steps away from true bubble sort though, which is IMO simpler.

    For i = collectionObject.count To 2 Step -1

        ' hasSwapped = False

        For j = 1 To i - 1

            If collectionObject(j) > collectionObject(j + 1) Then

                collectionObject.Add collectionObject(j), After:=j + 1
                collectionObject.Remove j
                ' hasSwapped = true

            End If

        Next j

        ' If Not hasSwapped Then goto sorted ' Exit For

    Next i

sorted:

In short, instead of bubbling up the same object each time, the BubbleSort drops the current item for the next one when it finds a larger item. This means after each inner loop the item at position i is in the correct place, which is why the outer loop is counting down not up.

hasSwapped uncommented allows for early exit if the sequence is already sorted. For more information on the bubble sort, wikipedia has a great article. I also found this site with some cool animations while trying to figure out your sorting algo.

Also note that AssignUnknown is no longer needed.

Error Handling

As we discussed in the comments, the comparison operators will not be defined for objects that do not have a default property. It can't be fixed, but we can raise a more descriptive error

    On Error Goto no_default_property

    ' bubble sort

sorted:
Exit Function
no_default_property:

    If Err.Number = 438 Then ' preferably use the vb constant that I don't know
        Err.Clear
        Err.Raise 438, "Sort", "An item in the collection does not have a default property"

    End If

End Function

After Thoughts

Consider writing CollectionToArray and ArrayToCollection functions so you don't need to duplicate sorting methods for Collections and Arrays. Also consider a Sorted function that returns a sorted copy.

Sub Sorted(collec As Collection) As Collection

    Set Sorted = collec 
    Sort Sorted

End Sub

I was tempted to insist you use a Swap function like this.

Sub Swap(ByRef a As Variant, ByRef b As Variant)

    Dim t as Variant
    t = a
    a = b
    b = t

End Sub

However, you are supporting objects in your collection, you will need to make that routine SwapUnknown that uses AssignUnknown. You could avoid calling IsObject(a) twice but I prefer the simpler solution.

Sub SwapUnkown(ByRef a As Variant, ByRef b As Variant)

    Dim t as Variant
    AssignUnknown t, a
    AssignUnknown a, b
    AssignUnknown b, t

End Sub

I didn't include this because:

• Your current swap method could be faster depending on how Collection is implemented
• Abstracting your current method seems pointless.
• It doesn't require AssignUnkown, which cuts out some ugly code.