So what does a MathOverflow moderator do?
Ideally, not much. The only task that moderators are responsible for is dealing with flags. When community members flag a post for moderator attention or when certain user behaviors trigger some automatic warnings, moderators get a notification. Dealing with such a flag usually takes a few seconds. The vast majority of flags are dismissed because the community is perfectly able of dealing with them and often have by the time moderators get around to it. Sometimes more situations arise where a more substantial response is necessary but it still takes very little time to deal with those.
The workload is shared among all moderators and is completely voluntary. When I have a spare minute, I visit the site and clear all the easiest flags. If I have another spare minute, I might clear some flags that require more substantial actions if there are any (and usually there aren't any). Otherwise, I will leave those for the next moderator who comes along. Serious flags are very rare and the whole processing time adds up to two minutes or so per day. I'm also one of the more active moderators. Other moderators prefer acting as backup when others have to spend time away from MathOverflow or when something highly unusual happens.
There are also a few maintenance tasks but these are very occasional and completely optional. I mostly do the tasks I enjoy doing, such as cleaning up the tag ecosystem, and I leave the tasks I find less enjoyable to other moderators. These maintenance task pop up very rarely but they can take more than a few minutes. A good technique to avoid burden is to ask community members to do these in their spare time. For example, I regularly post tag maintenance tasks on meta and the new review mechanism has made the task of deleting old off-topic questions almost entirely obsolete.
There are also some perks to being a moderator. It's a great synergistic activity and you get to meet a lot of interesting people. I've been having a lot of fun thinking about how various "Math 2.0" projects can cooperate with MathOverflow in some way or another. These side activities vary a great deal from moderator to moderator but the opportunities are not hard to find!