I have dealt with a lot of extremely dirty computers, and a part of a toothpick won't bother the inside of your case. The fans are probably the only things it will do anything to, and it will just be knocked away from it. I would say the worst case, Final Destination (the movies) type of problem that could happen is the tiny piece of wood absorbing moisture and causing a short on one of the printed circuit boards... haha, but that would be Hollywood movie worthy.
A desktop case fan is one of most easily replaced parts in a computer. You can get one for a few dollars, or even cheaper if you buy a used one. If it dies, you can probably get away with not having one as long as your CPU fan is in decent condition and not as furry as a yeti, especially if you don't have a performance graphics card installed. I know you didn't ask about this, but I figured it might interest you since it is so cheap. You would just have to turn your PC off, unscrew 0-4 screws (some computers use fans held in with clips), unplug one very easy wire and put the new fan in, following the same steps. Just measure the fan with a ruler in millimeters, or convert inches and round to 60, 80, 92 or 120mm. I would bet it is an 80mm fan though. Also, look for an arrow on the outside of the fan that points in the direction airflow, and aim it in the same way the last one was, most likely going out of the case.
Shipping will cost more than the fan itself, unless you buy a high-performance one or get free shipping. If the fan has a white piece of plastic at the end of the wire with 4 holes, which it most likely does, then it uses a molex connector. Total repair time is like 3 minutes. There is one on Amazon for 60 US cents. Spend a dollar more than that to drastically increase it's lifespan, and to get one that doesn't have an overly bright LED on it that can't be turned off.
Sorry, I know this is probably overkill, but I just joined the forum and want to contribute. Also, I love computers and just happened to be in the mood for doing this.