The problem is not in storing the hash in PHP itself; although a database is generally recommended.
The problem is using the hash as a plain-text password/secret; this is no different than any plain-text password. If someone was able to view the PHP source code they would have the hash and thus 'password'. Remember, password hashes are not secrets even though they should generally be treated confidentially1.
Instead, accept the username/password as plain-text - although, do use SSL to encrypt the password over the connection - and verify this against the hash2. At no point is a hash from the user trusted - as it cannot be proved that it was generated from an actual secret.
Once the submitted username/password has been validated on the server against the stored hash then a session nonce is established; it is this unguessable per-session secret that is then used to re-validate/authorize the user each subsequent request3. This is 'automatically handled' in PHP when creating a new session.
See the "Do's and Don'ts" and "Don't write your own password checker" answers, which are especially pertinent.
1 In real-world applications there should be a high priority given to confidentiality and preventing unauthorized disclosure. This is because most user choose stupidly short or common passwords (at least when given the chance) that can be brute-forced quickly. This can be mitigated by encouraging secure passphrase usage and secondary validators.
2 The newer password_verify
/ password_hash
(or an equivalent library / backport) functions should be used as these correctly handles basic details including
Applying a correctly generated salt which prevents rainbow table attacks, and
Using an appropriate (ie. bcrypt) hash function which mitigates brute-force attacks on strong passwords/passphrases.
(If planning on using SHA - don't! - for hashing passwords, stop and read the links..)
3 Unlike using a hash-as-a-password this is only susceptible to session-hijacking, even if the hash in the PHP source code were leaked: it is also unique per-session. The attacker needs access to the client's cookies (via local access or Cross-site scripting (XSS); XSS can be mitigated with HTTP-Only cookies) or a method of intercepting the HTTP request (which can be mitigated with HTTPS); then, within that session, an attacker could impersonate the authenticated client.
As a long aside: if there was a need to store data in cookie (or otherwise sent back with a server request) and to ensure that it came from the server to begin with, one solution would be to use an HMAC as an authentication code. But this does not apply here.