The first (A) is that TextSecure truncates SHA-2 hashes and, particularly, HMAC-SHA2 MACs. MAC truncation is extraordinarily common and is a design goal for most authenticated encryption schemes. So far as anyone knows, truncation of SHA-2 hashes and HMAC-SHA2 MACs is completely safe; that is, HMAC-SHA2 isn't somehow less safe when truncated than GCM or Poly1305.
The second (B) is that the authors believed TextSecure to have swapped some of the parameters for the HMAC algorithm. Moxie has asserted that this is a mistake by the paper authors, and that TextSecure uses HMAC conventionally.
The third (C, and most of the rest of the paper) is an identity misbinding attack. An identity misbinding attack happens when it becomes possible for Mallory to present Alice's public key to Bob instead of her own, and have Bob reply with a message encrypted to Mallory-seeming-Alice. Mallory can't decrypt the message, but because it's encrypted to a key Alice has, Alice might act based on the message contents. Identity misbinding problems are endemic to public-key cryptography. They are a real issue for protocols in which computers chat with each other and take action (like, authorizing an HTTP request) based on identities. They are less of an issue in chat protocols. Of particular importance, contextually: every other chat protocol has much, much worse key management problems.
Brian Smith says that this paper has value as part of a formal conversation about the security properties of TextSecure. I'm not qualified to evaluate that statement. I don't see anyone saying the paper has practical import for TextSecure or its users. TextSecure remains the best thing out there, I think.