The only thing I'd worry about is people that use it as a two-click solution and don't edit the question contents to make it appropriate for a non-chat setting.
In addition, it sends the entirely wrong messages to the people who abuse the chat by demanding instant answers in the chat and can't be bothered to take a few minutes to write up a well-phrased question on the main site, where it may help other people as well.
It teaches them to not give a crap, because someone clicks a button, someone else maybe turns the collection of single messages into a real question, and in the end they may even get answers without doing any work of their own. Seriously, I don't want those people.
I want people who understand that unconditionally demanding immediate answers from an awesome community without giving anything back is not the way it works.
So if someone starts posting their questions in the chat, ask them to take a moment and write a real question on the main site, maybe then linking to it in the chat room, so everybody benefits.
If they actually care to spend that time, it may turn out to be a good question, instead of a few chat messages buried somewhere – here's an example.
But if they can't be bothered, I have no intention to spoonfeed them.
At this very moment, there's a chat message on chat.SO containing "here's my error message: [...] how do I solve this?", flagged for moderator attention by the user who posted it, with the flagging message being
no one paying attention at my question
In my opinion, behavior like this is actively harmful to Stack Overflow and the community, and providing functionality that actually encourages this behavior really is the last thing I want to do.