A more readable InStr: StringContains
My 2 cents,

the first function seems fine, you could make it a little DRYer by just setting the compareMethod in your if statement and then have only 1 complicated line of logic. And if you are doing that, you might as well put the Cstr's there.

Public Function StringContains(haystack, needle, Optional ByVal caseSensitive As Boolean = False) As Boolean

   Dim compareMethod As Integer

    If caseSensitive Then
        compareMethod = vbBinaryCompare
    Else
        compareMethod = vbTextCompare
    End If
    'Have you thought about Null?
    StringContains = (InStr(1, CStr(haystack), CStr(needle), compareMethod) &lt;&gt; 0)

End Function


Notice as well that I love the idea of searching for needles in haystacks, I stole that from PHP.

For StringContainsAny, you are not using the code you wrote for StringContains, you repeat it. If you were to re-use the first function, you could do this:

Public Function StringContainsAny(haystack, ByVal caseSensitive As Boolean, ParamArray needles()) As Boolean

    Dim i As Integer

    For i = LBound(needles) To UBound(needles)
        If StringContains(CStr(haystack), CStr(needles(i)), caseSensitive) Then
          StringContainsAny = True
          Exit Function
        End If
    Next

    StringContainsAny = False 'Not really necessary, default is False..

End Function


For the last one I wanted to you consider passing values that you will convert as ByVal, since you are going to make a copy anyway of that variable.

Public Function StringMatchesAny(ByVal string_source, ParamArray potential_matches()) As Boolean

  string_source = CStr(string_source)

  ... 'That code taught me a new trick ;)

End Function