Encryption with "constant" initialization vector considered harmful
For block ciphers, it depends on which mode of operation you're using &mdash; nobody uses just a plain block cipher for anything, at least not unless all their messages are shorter than a single cipher block (typically 8 or 16 bytes).


ECB mode, which just amounts to chopping the message up into blocks and feeding each block through the cipher, does not use an IV.  As you correctly note, it's also insecure: it leaks information about duplicate blocks in the plaintext.
OFB and CTR modes turn the block cipher into a synchronous stream cipher.  Thus, they have the same security issues as other stream ciphers: if the same key+IV combination is used for two messages, an adversary that captures both can take their difference to eliminate the keystream, leaving only the difference of the two plaintexts (which is often enough to recover at least parts of both of them).  It gets even worse if they can capture more than two messages using the same key+IV.
The CBC and CFB modes are somewhat more resilient to simple IV reuse.  As with all other common operating modes, though, using the same key+IV to encrypt two messages sharing a common prefix (of one or more blocks) will reveal the presence and length of that prefix, since the ciphertexts will be identical up to the point where the first plaintext difference occurs.  CFB mode also leaks the difference of the first non-identical plaintext blocks in the two messages.