Turbine owns or licenses several trademarks (words, phrases, or graphic depictions) used in conjunction with operating and selling the game service. In addition, imagery, sounds, code, and data from the game itself are covered by the game's copyright.
If you wish to use any of these things for your own published work (especially if there will be money involved), you should contact Turbine/WB's legal department and ask about licensing opportunities.
But if you're talking about something simple like fan art for yourself and your friends, most gaming companies are quite happy for their fans to create and freely distribute limited creative works based on their games. Just don't give the impression that your work is endorsed by the company unless it actually is. As long as there's no money involved and your work portrays the game in a positive light, you're unlikely to encounter any issues.