How to check success status of opened URL in Tasker?
You could use an "HTTP Get" action to query the site.  The response code sent back by the website is stored in the %HTTPR variable.  So %HTTPR = 200 means the site loaded sucessfully.  You can then make the rest of your task conditional on that variable, eg. IF %HTTPR = 200 THEN Go Home (or however you are "closing" Dolphin).  I know that %HTTPR is dependent on the mime type of the HTTP Get request so you'll have to tweak that a bit, and you may choose to look for different response codes to trigger different actions. 

There's probably other ways to do it but it would depend on the site you are trying to reach.