It really depends on how big the server is. If it's small, you may be able to get away with using premium web hosting, however, if it's a large server (Maybe 50+ people) you WILL need a dedicated game hosting server or VDS. Web hosting providers usually have a monthly data transfer limit (Bandwidth usage) and a large server will quickly eat up that quota.
You also have to look at reliability. Gaming VDSs and physical dedicated servers have much higher reliability and down-time plus they are meant for the kind of bandwidth usage you see from hosting video games.
In addition to reliability, you need a lot of processing power to handle hosting a game server, especially if there are a lot of people playing. Web hosting services generally split a single server amongst possibly 30+ clients, this means, while the server can handle heavy loads, you will not have the server's full power. With a VPS, you can choose the power you need and you can, for a little more money, get unmetered hosting (Unlimited bandwidth). A dedicated server would be the best though. The other downside of web hosting is it's very hard to find one that will let you run your own daemons (Services) without paying premium prices. This is usually where VDSs come in.
And finally, hosts often check with humans to find servers running anything illegal or could get them in trouble. So you'll need to find a host that allows people to host stuff like WoW emulators which are few and far between with US-based hosts.
If you were to find a web space host that could handle a server and allows you to run daemons, I can almost guarantee they'll shut you down anyway since you'll be killing the server's bandwidth with a game rather than a website.
