This will be dangerous if there is a high probability of repeated initial blocks across plaintext. Cryptographers like to make as few assumptions as possible about the structure of plaintext and so are very conservative in their designs and recommended usage.
Two examples of families of plaintexts where CBC with fixed IV might be bad are as follows:-
Consider a classified organisation where messages are prefixed with header either "TOP SECRET - FOR YOUR EYES ONLY" or "CONFIDENTIAL - CIRCULATE". If these are encoded into ASCII and encrypted with a 128-bit block-size cipher in CBC mode with fixed IV, adversaries will be able to group together the messages with the same classification.
Consider an automated update of a stock portfolio that doesn't trade much. Perhaps a message might read:
With CBC with fixed IV an adversary could see multiple identical messages encrypted with the same key, but when pending transaction does occur, they would be able to detect it and also make a strong guess as to the stock likely to be traded.
There are also more active approaches that a cryptographer seeks to avoid. They will consider a scenario where the adversary is able to control part of the ciphertext (e.g. if a message begins "You wrote: '
<CONTENT OF LAST RECEIVED MESSAGE>'". If an adversary submits a message where the fixed prefix is one byte short of a block size, and then submits 256 messages with the same prefix appended with all possible follow on bytes, they recover the first byte of user plaintext from the first message. This is similar to the real-world ChopChop attack on WEP.
It is certainly possible to specify that plaintext does not have these sort of structures in order to avoid these attacks, but security professionals feel that the defensive burden should fall to the cryptographer rather than the user.