Those insects that look like moving dots on your indoor palm tree could be mites -- in particular, spider mites that are often an issue with palms. Although not all species leave fine webbing behind, most spider mites do. A small mite infestation is not usually a problem, but in large numbers, they can eventually cause leaves to turn yellow and drop. Several control methods are effective on mites, and if you incorporate more than one, your indoor palm tree will continue to thrive.
Move the potted palm outdoors, or into a bathtub or shower. Spray the stems, trunk, and tops and bottoms of the fronds with a steady stream of water from a garden hose or handheld shower head to dislodge and kill the mites. You can gently shake or pat the plant dry with a towel, if desired, before moving it back to its regular location. Repeat this washing once a week until the mites are under control, then repeat once a month to keep them at bay.
Spray the palm to wet the stems, and the tops and bottoms of the fronds with a ready-to-use insecticidal soap or horticultural oil labeled to kill mites. Repeat one or two times a week for three weeks to completely kill all the mites infesting the palm. Test the product on an inconspicuous leaf first about three or four days before applying it to the whole plant to ensure it does not discolor or dry out the leaves. Protect curtains, rugs and other fabrics with a sheet or plastic cover so the soap or oil does not damage them.
Dust palms regularly to create an environment that is less conducive to mites. Also, keep the potted palm well watered so it is not drought stressed. Increase the humidity around the palm, such as with a humidifier. Spider mites prefer hot, dry and dusty locations.