How do I turn off Oracle password expiration?

To alter the password expiry policy for a certain user profile in Oracle first check which profile the user is using:

select profile from DBA_USERS where username = '<username>';

Then you can change the limit to never expire using:

alter profile <profile_name> limit password_life_time UNLIMITED;

If you want to previously check the limit you may use:

select resource_name,limit from dba_profiles where profile='<profile_name>';

As the other answers state, changing the user's profile (e.g. the 'DEFAULT' profile) appropriately will lead to passwords, that once set, will never expire.

However, as one commenter points out, passwords set under the profile's old values may already be expired, and (if after the profile's specified grace period) the account locked.

The solution for expired passwords with locked accounts (as provided in an answering comment) is to use one version of the ALTER USER command:

ALTER USER xyz_user ACCOUNT UNLOCK;

However the unlock command only works for accounts where the account is actually locked, but not for those accounts that are in the grace period, i.e. where the password is expired but the account is not yet locked. For these accounts the password must be reset with another version of the ALTER USER command:

ALTER USER xyz_user IDENTIFIED BY new_password;

Below is a little SQL*Plus script that a privileged user (e.g. user 'SYS') can use to reset a user's password to the current existing hashed value stored in the database.

EDIT: Older versions of Oracle store the password or password-hash in the pword column, newer versions of Oracle store the password-hash in the spare4 column. Script below changed to collect the pword and spare4 columns, but to use the spare4 column to reset the user's account; modify as needed.

REM Tell SQL*Plus to show before and after versions of variable substitutions.
SET VERIFY ON
SHOW VERIFY

REM Tell SQL*Plus to use the ampersand '&' to indicate variables in substitution/expansion.
SET DEFINE '&'
SHOW DEFINE

REM Specify in a SQL*Plus variable the account to 'reset'.
REM Note that user names are case sensitive in recent versions of Oracle.
REM DEFINE USER_NAME = 'xyz_user'

REM Show the status of the account before reset.
SELECT
  ACCOUNT_STATUS,
  TO_CHAR(LOCK_DATE, 'YYYY-MM-DD HH24:MI:SS') AS LOCK_DATE,
  TO_CHAR(EXPIRY_DATE, 'YYYY-MM-DD HH24:MI:SS') AS EXPIRY_DATE
FROM
  DBA_USERS
WHERE
  USERNAME = '&USER_NAME';

REM Create SQL*Plus variable to hold the existing values of the password and spare4 columns.
DEFINE OLD_SPARE4 = ""
DEFINE OLD_PASSWORD = ""

REM Tell SQL*Plus where to store the values to be selected with SQL.
REM Note that the password hash value is stored in spare4 column in recent versions of Oracle,
REM   and in the password column in older versions of Oracle.
COLUMN SPARE4HASH NEW_VALUE OLD_SPARE4
COLUMN PWORDHASH NEW_VALUE OLD_PASSWORD

REM Select the old spare4 and password columns as delimited strings 
SELECT 
  '''' || SPARE4 || '''' AS SPARE4HASH,
  '''' || PASSWORD || '''' AS PWORDHASH
FROM 
  SYS.USER$ 
WHERE 
  NAME = '&USER_NAME';

REM Show the contents of the SQL*Plus variables
DEFINE OLD_SPARE4
DEFINE OLD_PASSWORD

REM Reset the password - Older versions of Oracle (e.g. Oracle 10g and older) 
REM ALTER USER &USER_NAME IDENTIFIED BY VALUES &OLD_PASSWORD;

REM Reset the password - Newer versions of Oracle (e.g. Oracle 11g and newer) 
ALTER USER &USER_NAME IDENTIFIED BY VALUES &OLD_SPARE4;

REM Show the status of the account after reset
SELECT
  ACCOUNT_STATUS,
  TO_CHAR(LOCK_DATE, 'YYYY-MM-DD HH24:MI:SS') AS LOCK_DATE,
  TO_CHAR(EXPIRY_DATE, 'YYYY-MM-DD HH24:MI:SS') AS EXPIRY_DATE
FROM
  DBA_USERS
WHERE
  USERNAME = '&USER_NAME';

For development you can disable password policy if no other profile was set (i.e. disable password expiration in default one):

ALTER PROFILE "DEFAULT" LIMIT PASSWORD_VERIFY_FUNCTION NULL;

Then, reset password and unlock user account. It should never expire again:

alter user user_name identified by new_password account unlock;

I believe that the password expiration behavior, by default, is to never expire. However, you could set up a profile for your dev user set and set the PASSWORD_LIFE_TIME. See the orafaq for more details. You can see here for an example of one person's perspective and usage.