Is there a very light material that could deorbit without burning up? I am thinking of a very light, strong, heat resistant thread, rope, net, or sheet that sheds heat so fast as it deorbits that it doesn't melt or burn.
Silicon Carbide (SiC) is close to material you are searching for. It's slightly denser than aluminium 3.16g/cm3 but lighter than titanium. It doesn't melt under any conditions, it sublimes at 2,830 °C and has excellent thermal conductivity. Hexagonal form has better thermal conductivity than copper and silver 490 W/mK. It's very resistant to oxidation and is used for production of heating elements especially for melting glass. It can bi made in fine SiC fibers with diameter 50-150 micrometers. This fibers are resistant to strong heating and used in filament pyrometry for measuring gas temperatures up to 2500K.
Second interesting material is Boron Nitride (BN) - density 2.1 g/cm3 - lighter than aluminium, with excellent thermal conductivity 751 W/mK. The theoretical thermal conductivity of hexagonal boron nitride nanoribbons is in the range of 1700–2000 W/m·K. Material has high melting point 2,973 °C and it sublimes in high extent. It can be made in the form of nano tubes. It has lower resistivity to oxidation than SiC, but bulk material can hold up to 850°C, BN aerogels can resist up to 1200°C. BN nanotubes are generally considered flame resistant.
Interesting article how different materials behave during atmospheric reentry.