Voted no, and I will try and respond in segments to make what I am going to say clear. 
  
"To be honest, it can get quite annoying answering the same questions 50 times in a day." 
  
The questions people usually ask in Live Chat are simple and basic and usually only take a few seconds to answer. And by answering these questions you may have just relived the person of confusion and also possibly aggravation with the client that they have been having for hours or even days, and trust me they appreciate your assistance. I can not recall anyone taking help for granted. 
  
"This will also help prevent ban evaders/Website advertisers from making accounts to advertise in the chat room." 
  
To the second part, I can see your point about ban evaders. But honestly, usually ban evaders are people who are interested in the market and try to scam a second time, so they would be posting around the forums anyways to get back into the market so a limit to get into the Live Chat would be futile. And about website advertisers in the Live Chat, if they are ever a problem. Usually there is a Mod in the chat in order to kick/ban them from the Chat so they can not return. 
  
Hope that makes since, interested idea though I love hearing new suggestions.