Yes, you can tape a furnace air filter to a box fan and use it as an air purifier. Here’s a long article with many links and videos on the options. You have to be careful – my own family has burned out fan motors trying to put seemingly light weight nets over them to keep childrens’ fingers out (no smoke, no fire, they just quit working, but still a good idea to switch fans off when you are away from home as a general principle). You can in fact just tape the filter to the back of the fan and be done. More elaborate setups build a cube out of the filters with the fan inside, increasing the surface area and putting less stress on the motor. Some approaches use cardboard cutouts to channel the air flow (which seems harder on the motor to me). People are Covid-focused these days, but seasonal allergies will be around long after Covid is gone. This seems like it could be a particularly cheap and effective intervention in offices and schools.

Modern, cheap plastic fan blades won’t cut off or even break your fingers by the way, in my experience, and I do have experience. As a relatively tall man, I also have experience being whacked in the head by a ceiling fan blade a time or two. Both these things cause significant pain, but I am still here with all my body parts and no long-term damage.