Could be a leaky seal somewhere. You should get underneath the car (put it on ramps; DON'T trust a jack by itself), take off the plastics underneath, and see if the oil pan/lower part of the engine has oil residue on it somewhere. Also check around the water pump area; if the pump is going bad it could be leaking oil.
Can anyone comment about whether the belt tensioner can develop a leak (I don't have any experience with that)?
If it turns out to be a leaky seal, you can probably let it go for while; just keep your oil level up. A leak from the water pump or tensioner should get fixed right away (pump could be going bad). If its just burning oil, you can try running some engine cleaner (Seafoam, or an equivalent) through it and see if that helps. Otherwise, if its just burning oil there's not much you can do, except have some rebuilding work done, or put in a new engine. You can live with it burning oil for a while; might have a problem passing emissions inspection though.