I would imagine with the amount of visitors to LJ the board is probably hosted on a VPS or dedicated server and both can be expensive.
My wife has so many business clients that she also host and websites she maintains that she has her own dedicated servers and owns her own hosting company. It cost her several 100 bucks a month for her servers not to mention the upkeep and support for them. I've seen her servers bills go well over 1K a month. She doesn't allow any ads on any of her client sites, nor her hosting company because her clients pay her to host them without ads.
Her servers are located in a top US datacenter in Chicago with colocation servers in Atlanta, LA, and Dallas.
The more visitors to a website the more bandwidth and ram it uses and to keep a high traffic site online like LJ it can get expensive.
According to the info for LJ that is public LJ is hosted with
http://webnx.com/
They look to be a hosting company for high volume websites not the average website on a shared hosting server like I'm sure most of you have.
My free plans site takes an average of 1500 unique hits a day and is hosted on a reseller hosting account I own which is shared hosting.
I don't pay any attention to the ads on LJ because I understand it can be expensive to maintain and support. One thing that will make me haul butt is a popup ad on a site. I leave those never to return.
I'm sure with the amount of site visitors, blogs, pictures, etc it cost a good bit of money to maintain and manage LJ.