Lets suppose we have a communication protocol given where messages $(1)$ and $(2)$ are send over an unsecure network.
There are some prerequisites like distributed certificates etc., but they should not matter for this question).
Message (1) is send as plaintext from $A$ to $B$ and does contain the identity of $A$. Party $B$ responds with a signature over the received identity and some other data (e.g. a public DH key) and sends it over to $A$.
A protocol that starts like this design allows an attacker to get a lot of signatures (as much as needed) and the input is even something under his control (the identity of $A$).
In my opinion this may open door to a lot of attacks the attacker can use. I don't know any but I can imagine that have been some and this kind of design should always be avoided.
I am open for any kind of comment on this. Especially if there were some relevant attacks in the past.