I'd use a 'where not exists' -- exactly as you suggest in your title:
SELECT `voter`.`ID`, `voter`.`Last_Name`, `voter`.`First_Name`,
`voter`.`Middle_Name`, `voter`.`Age`, `voter`.`Sex`,
`voter`.`Party`, `voter`.`Demo`, `voter`.`PV`,
`household`.`Address`, `household`.`City`, `household`.`Zip`
FROM (`voter`)
JOIN `household` ON `voter`.`House_ID`=`household`.`id`
WHERE `CT` = '5'
AND `Precnum` = 'CTY3'
AND `Last_Name` LIKE '%Cumbee%'
AND `First_Name` LIKE '%John%'
AND NOT EXISTS (
SELECT * FROM `elimination`
WHERE `elimination`.`voter_id` = `voter`.`ID`
)
ORDER BY `Last_Name` ASC
LIMIT 30
That may be marginally faster than doing a left join (of course, depending on your indexes, cardinality of your tables, etc), and is almost certainly much faster than using IN.