No, they don't. Most forums have report buttons.
For example, here is from AnimeBytes' forums (the only other forum I go on):
We could choose to bring on 300 moderators like MAL does, but then we would probably end up with some really fucking awful moderation like MAL did. We chose quality over quantity.
Think of us like the Fire Department: you call 911 (or whatever the equivalent in your nation is) and we show up. We don't patrol the town looking for fires. Ambulances too: they don't patrol, they just wait for a call.
Actually, a better analogy is the police: you call 911 (or equivalent) and we show up, but we're usually patrolling somewhere nearby. Many of us actually browse the forums quite regularly and act when we see things which need our attention, but we still can be summoned.
It's not our responsibility to be everywhere at once. It's our job to handle things once we receive a call and act if we see anything ourselves.
If you don't feel like it needs moderation, don't flag the post, and likewise don't complain when it isn't moderated (because we likely agree that it doesn't need this.
I don't know where you got this idea. I've run a fair number of mod teams (mostly for chats, which require more attention — but also for forums) and NEVER seen a mod team monitor every post 24/7. EVER.