The OpenPGP message format describes many "messages" which are the serialization of structures involving cryptographic algorithms. Some of those messages are meant to be exchanged as emails; when you send a "signed email", you must use your private key. The resulting message (the one which goes to the SMTP server) is not impacted by how the said private key is formatted when stored on a non-transient medium. In that sense, you can "use OpenPGP" (send and receive signed and encrypted emails) without having to store a "password-protected" private key; what OpenPGP requires is that a use can "own" a private key and "remember" it for non-trivial amounts of time. The private key itself should be stored with a high level of confidentiality, since such confidentiality is crucial for the whole security of the scheme. The user private key is often stored in a file, encrypted with a symmetric key derived from a user password ("passphrase" in PGP-speak). Less often, but equally valid, could the private key be stored in a tamper-resistant device such as a smartcard, and accessed through a dedicated API such as PKCS#11; support for PKCS#11 exists for GnuPG. How the device stores the key is up to the device but rarely involves encryption with a password-derived key.
OpenPGP also defines message formats for storing a serialized private key, in particular with symmetric encryption using a key derived from a password. Therefore, when an OpenPGP implementation wishes to store a private key as a file with password protection, it tends to use that specific message format. Using this format allows for transferring private keys ("key rings") between distinct implementations.
So while password-protection of private key is not a mandatory requirement for claiming "OpenPGP support" as far as on-the-wire messages are concerned, it is still recommended, and there is an "OpenPGP-approved" way of doing that password protection, which has portability benefits and is followed by actual implementations.