Why don't paper cups burn when they hold hot drinks?
The short answer
A paper cup can hold a hot drink because the drink does not heat the paper to the temperature needed for it to catch fire.
Hot is not the same as hot enough to ignite
At ordinary air pressure, water boils at about 100°C (212°F). The Royal Society of Chemistry gives about 230°C (446°F) as an approximate ignition temperature for paper in its teaching notes. The exact temperature depends on the paper and the conditions, but the gap helps explain why a hot drink does not normally ignite a cup.
Heat does pass from the drink into the cup, so the cup can become hot. But warming paper is not the same as heating it until it catches fire.
What if a flame heats a cup of water?
In a demonstration, a paper cup containing water can be heated from below without the part touching the water burning. Heat passes through the paper into the water. Water takes a lot of energy to warm up, and movement in the water spreads heat away from the hottest area. While liquid water remains, it stays close to its boiling point instead of getting much hotter at ordinary air pressure.
Paper above the water can still burn
The rim, folded parts, or other paper above the water may char or catch fire because water is not carrying heat away from those areas. The cup can also burn after the water is gone. A paper cup is not fireproof and is not made for use over a flame.
A cup that does not burn can still be hot
The cup may feel hot to your hand even though it is not burning. Heat from the drink can pass through the cup, so handle hot drinks carefully.