It wouldn't, necessarily -- I thought about this a little, and there are ways to identify commonly used passwords without storing them in plain text.
They could check passwords on login (before salting/hashing) against a blacklist of 'commonly used passwords'. This is probably the most secure method, as it only implies that users passwords are not on the blacklist, but does not imply plain text storage or unsalted hashes.
If they were not salting their hashes, then they could query their database for hashes that match the hashed version of commonly used passwords. Similar to the blacklist above, but implies that the passwords are stored unsalted, which is sad.
They could check passwords on login (before salting/hashing) against a blacklist of 'commonly used passwords'. This is probably the most secure method, as it only implies that users passwords are not on the blacklist, but does not imply plain text storage or unsalted hashes.
If they were not salting their hashes, then they could query their database for hashes that match the hashed version of commonly used passwords. Similar to the blacklist above, but implies that the passwords are stored unsalted, which is sad.