You can block the IP address entirely by making an edit to WebLink's web.config file. Instructions on how to do so are found here. Note that this is more of a general IIS configuration rather than a Laserfiche specific one.

Another thing you could try is disabling the keepalive calls in the doc viewer. WebLink's doc viewer has javascript that keeps the session active while the user is viewing public documents. The original idea was they could open something, go away for lunch, come back an hour later and still have access to the document they were viewing. If you disable this timer with some custom javascript, the session will timeout in the normal 20 minutes regardless of whether the user is looking at a document or not. Instructions on how to make this customization can be found here.

Other than that, there aren't really any options for limiting the number of connections used to a certain amount. Unless the user is acting maliciously, i would think making the javascript edit would at least ease the situation. If the user is acting maliciously, you'll probably want to block that user anyway.