A toilet paper clog in your toilet is unusual, because toilet paper is designed to break down in water. However, if you stuff an unusual amount of toilet paper in the toilet at once, it might get caught in the internal trap -- especially if your toilet dates to the mid-1990s, when manufacturers hadn't perfected the design of low-flow toilets and made the traps too small. The paper in the front of the clog might break down slowly while that in the back remains intact because it is out of water contact. Any drain cleaner that dissolves hair and grease can help toilet paper dissolve, but that doesn't mean you should use one in your toilet.
Drain Cleaner? Not a Good Idea
Most drain cleaners are either acidic or alkaline, and the farther from pH-neutral a cleaner is, the faster it dissolves obstructions like toilet paper. Drain cleaners containing sulfuric acid and -- on the opposite end of the pH scale -- sodium hydroxide will probably make short work of toilet paper, but they are harmful for your toilet pipes and the environment, and they are especially harmful for septic systems. They are also dangerous for you -- pouring a strong drain cleaner in a toilet with standing water creates a bowlful of hazardous material that can burn the skin of anyone who tries to clear the toilet the right way.
The Right Way
Every house with a bathroom needs two handy tools in the utility closet. The first is a toilet plunger and the other is a toilet auger. The chances of a toilet paper clog resisting either of these tools are miniscule.
Safe Chemical Options
Enzyme drain cleaners
Enzyme cleaners contain catalysts that speed the breakdown of waste with microorganisms in the water, and these microorganisms are undoubtedly happy to feed on toilet paper. The enzymes are nontoxic and safe for pipes and septic systems, but they take a long time to work. If you're trying to clear a clog in a toilet that you can leave unused for at least 24 hours, an enzyme cleaner may be the right choice for you.
Dish soap sinks to the bottom of the bowl lubricating the trap and the paper and allowing water to seep through and eventually open the drain. Pour a generous amount in the toilet, wait for several minutes, then pour in a bucket of hot water. Don't use boiling water, which can crack a cold toilet bowl.
Vinegar and baking soda
This mixture is often touted as a natural, safe alternative to harsh chemical drain cleaners; when mixed together, vinegar and baking soda release carbon dioxide, which turns the mixture in a foaming volcano that is hard for toilet paper to resist. Pour 1/2 cup baking soda into the toilet and follow this with 1/2 cup of vinegar, then close the toilet, because the mixture will make bubbles in the standing water. Wait 5 or 10 minutes, and if the water doesn't drain, repeat the treatment.