Those tiny orange spiders in your plants are not spiders, but the two-spotted spider mite. They need to be removed immediately, or they may kill the plants. Spider mites feed by repeatedly piercing plants and sucking out the contents. Even worse, they reproduce rapidly. Like spiders, spider mites create silk web-like threads.
Spider mites have oval bodies and eight legs. They average less than 1/20th of an inch long. Two-spotted spider mites have two dark spots on their backs. These spots are accumulations of bodily wastes, so no spots may appear on mites that have shed old skins. Many species of spider mites become bright yellow, orange or red as winter approaches.
Leaves of plants infected by spider mites become covered in yellow, grey or bronze dots. The leaves eventually wither and fall off prematurely. Damage to leaves is worse if there has been a drought, or if the houseplant has not been regularly watered. A light cover of fluff on the plants may be the mite's webs.
Many creatures eat spider mites, including ladybugs, minute pirate bugs, lacewings and other, smaller species of mites. If they show up on houseplants, run the plant quickly under running water in a sink at least once a week. Use a damp cloth to wipe the leaves of larger house plants that cannot be moved to a sink or bathtub. Hose trees and other outdoor plants with plain water to remove and drown mites.
Some strains of spider mites are resistant to many insecticides. No insecticide can kill the eggs, so any insecticide must be reapplied 10 to 14 days after the first treatment to eliminate hatchlings. Neem oil or petroleum-based insecticidal oil can kill spider mites. Never apply these oils when the temperature is over 90 degrees Fahrenheit as the plant is too stressed to handle the oils. Spray or wipe the oil on the underside of leaves, where the mites congregate. The oil must get directly on them to be effective.