For these types of sites, nothing. They're serving mostly static content utilizing some sort of content management system to do so. It's not like it's a custom job with a lot of heavy processes, I/O, or even many database reads/writes.
I've been a professional web developer/programmer and internet marketer for 15 years, and now I'm an executive at a cloud hosting company, so I do have some idea of what I'm talking about.
Now I don't know if the fundraising call is for just the website or other associated organizational costs. I couldn't really tell from the posted article. Typically with these types of fundraising calls, the money isn't really for the hosting costs. If it is then they really don't know what they're doing and they should probably hit me up. Usually it's because they want to earn a full time salary off the website, or they have others who are earning a salary off the website as as such the "expenses" are high.
By the way, all this nonsense about google cutting CPC rates on liberty/anti-state websites and/or rankings is pure hogwash. That is not how it works. Usually if a website with this kind of traffic can't be monetized it's the fault of the website owner. I can't tell you how many times I've bought websites that make nothing and implemented a few small ad changes resulting in those same websites earning thousands/month. Maybe they should seek some help/advice in regards to that. The fact is that if their ad space is worth anything, then they'll get paid well for it. If Lew really believes that google is screwing him over then he should sell advertising directly to media buyers. There are plenty of them out there who will scoop up every single impression you have if its profitable on their end.
Just a random guess as I obviously haven't seen any of his metrics or demographics, but it's a strong possibility that he's seeing a lot of revenue lost due to the decline in desktop web browsing and the move into mobile. The monetization/advertising game does change quite a bit in the mobile space.
I'm not the biggest fan of the LRC these days, but I do hope they remain online. So I wish them luck in their fundraising.