No amount of filtering will rid your pool of "the green." You are not properly chlorinating your pool, which is allowing algae to grow. Filters do not control algae, proper chemistry does. And tabs are the devil. Yes, they add chlorine, but they also add CYA (cyanuric acid). CYA helps protect your chlorine level, until you allow the CYA to get too high, then you cannot add enough chlorine to actually chlorinate your pool. This is why you end each summer with green water. Then you exchange a bunch of water, which helps reset your CYA level, and you start over each year in the Spring. It's an endless cycle, which, as you've learned, is not sustainable.
If I may, you're running your pool without really understanding how all the equipment and chemicals work. Which is EXACTLY how we all used to do, until we found TFP!! You've come to the right place. As
@wireform has steered you, you need a proper test kit, and then you need to study up a bit about what we teach here. Once you get a handle on your chemistry, you'll be algae free!
Then you'll run your filter enough to keep the water crystal clear and you'll never see "the green" again (none of us do)! To answer your original question, the only real downside of running too large a pump though too large a filter is cost, your energy cost. But once you get your chemistry under control, you'll only need to run your pump a few hours a day (TBD, depending on several factors, like how much crud falls in your pool each day, how many swimmers, etc).
Start with the two links
@wireform gave you. Buy one of our recommended kits right away, and study up a bit on our method. Don't skimp on the test kit, it'll pay for itself right quick. Don't skip the lessons, either. In no time at all we'll have you straightened out, and you'll be amazed with how clean and clear your water can be.
Welcome to TFP!