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I want to update a table with consecutive numbering starting with 1. The update has a where clause so only results that meet the clause will be renumbered. Can I accomplish this efficiently without using a temp table?

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This probably depends on your database, but here is a solution for MySQL 5 that involves using a variable:

SET @a:=0;
UPDATE table SET field=@a:=@a+1 WHERE whatever='whatever' ORDER BY field2,field3

You should probably edit your question and indicate which database you're using however.

Edit: I found a solution utilizing T-SQL for SQL Server. It's very similar to the MySQL method:

DECLARE @myVar int
SET @myVar = 0

UPDATE
  myTable
SET
  @myvar = myField = @myVar + 1

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