Is it necessary to replace this cursor in SQL Server 2005?
So, you can simplify the CTE with some SQL magic:

WITH RecordsBeingUpdated AS 
( 
    SELECT 
        NOTE_COUNTER,
        MEMBER_UPI, 
        CASE_SEQ, 
        DIFF_ENCOUNTER_DATE, 
        ROW_NUMBER() OVER (PARTITION BY MEMBER_UPI, CASE_SEQ ORDER BY DIFF_ENCOUNTER_DATE ASC) AS RowNum 
    FROM 
        #TMP_NOTES   
    WHERE 
        TOTAL_CALLS &gt; 1       
)  

UPDATE RecordsBeingUpdated
SET    NOTE_COUNTER = RowNum 


Essentially, the virtual table built for the CTE has direct reference back to the concrete tables in the selection. This means you can update the CTE directly and SQL will update the corresponding concrete table within, provided the target table is the FROM table (not a joined) and there is no chance of ambiguity in what needs to be updated.

(Removed DIFF_ENCOUNTER_DATE from partition per comments).