まず、適切な ANSI 結合構文を使用します。
SELECT distinct usr.user_id
FROM OCN.users AS usr
inner join OCN.users_groups_subscribe AS usr_grp on usr_grp.subscribe_user_id = usr.user_id
inner join OCN.tests_users AS test_usr on test_usr.test_user_user_id = usr.user_id
WHERE
test_usr.test_user_test_id=116 AND
(usr.user_first_name LIKE '%ajay%' OR usr.user_last_name LIKE '%ajay%')
ORDER BY usr.user_first_name, usr.user_last_name
次に、結合を 1 つずつ (対応する列と一緒に) コメントしてみて、次のように行が取得されない理由を確認してください。
SELECT distinct usr.user_id
FROM OCN.users AS usr
--inner join OCN.users_groups_subscribe AS usr_grp on usr_grp.subscribe_user_id = usr.user_id
inner join OCN.tests_users AS test_usr on test_usr.test_user_user_id = usr.user_id
WHERE
test_usr.test_user_test_id=116 AND
(usr.user_first_name LIKE '%ajay%' OR usr.user_last_name LIKE '%ajay%')
ORDER BY usr.user_first_name, usr.user_last_name