Recently, I went to Legendary ConFusion, and attempted to sit in on as many writing-oriented panels as I could. I'm working through my notes and trying to share some of what I learned. Enjoy!
The short answer is: yes, your world really does need an economy.
It's also important to ask yourself, is it important to me to write a tight, well-written economy? as well as, what will the market bear? The amount of worldbuilding that may go into a hard sci-fi novel may be vastly different than what a YA novel set in the suburbs near Denver will require. You, the author, should have a pretty good grasp on what's going on in your world, but all ideas are subject to plot and pacing. You might know very minute details that won't make it onto the page. In fact, most economies are implied. You'll do far more work on your end, because it's really the results of your economy that are shared. Know when to say that's enough with the details, because your story should be, first and foremost, entertaining. Worldbuilding is important, but if you have a great setting lack compelling characters, no one will care.
Your world's economy drives the stakes of the story. Without money to be made, people won't stay in an area. Is your story set in the remnants of what was once a booming industry? A thriving trade hub? Your world's economic status will have had some effect on how your characters were raised, and perhaps what their view of the world is. (Think of Katniss' reaction to the Capitol, for instance) Are your people dependent on other countries to get the resources they need, or are they self-sufficient?
The geography of the world will dictate what kind of jobs are available, as well. A trade city will need to be accessible, not stuck at the peak of the mountains. You probably won't find many fisherman or farmers in the desert, but a jeweler would operate best in a thriving city. Think back to who settled your cities - the first successful settlers, in the long run, tend to become the people with influence and resources there. What drove people to settle there?
Economy isn't limited to money, but includes bartering goods and services - so long as people with valuable skills are around. What does your character offer to those around them? What would happen if your character came from a bartering community and found themselves in a currency based system - or the opposite?
Economy matters to everyone. The status of the economy determines whether (and what) a family eats, whether a country can afford to go to war. Economies can drive acts of desperation or lull them into complacency. Even the little guy can have more power than a king if they have enough resources to pool together.
While we tend to think of economy in terms of money, it can also come in the form of social exchanges, such as gender bias, or marriage for alliances. In our society, our careers are a form of economy. We suss out our standing in society and who our advantageous acquaintances are with the question what do you do?
Also, if you want to be 'outside the box' with your economic system and draw attention to it (which is totally cool!), try to make it strange enough to be interesting, but recognizable enough to sustain interest and not scare away readers.
Hopefully you found some decent food for thought in this. I know that for me, world building is something that I love in concept, but struggle with knowing what questions to ask. After sitting through this panel, I feel like I have a much better grasp on how to shape an economy.