There are 17 genera of palm trees grouped under the family name Arecaceae. Although these palms differ in height, growing habit and specific U.S. Department of Agriculture plant hardiness zone range, they are warm-weather loving plants that are intolerant of frost. Generally they grow a bunch of fronds on top of a long stem, which can be either smooth or textured. Palms differ in their water requirements, but most like sites in full sun. Some are salt tolerant, while some are not.
Palms, once grown, are very tough plants that likely will not succumb because of a single application of Roundup. However, repeated applications can cause damage to palm tree tissue that may take several months to recover from. Applying it to small seedlings can definitely have damaging effects, up to and including death. Especially during the growing season, plants are susceptible to herbicides, which damage their sensitive leaves and stems.
Roundup enters plant tissues and moves through the plant, inhibiting the production of amino acids and starving it. After about a week, and possibly after only one application, the plant will die. Broad-spectrum herbicides like Roundup can cause damage to most garden plants, especially if applied while wind is blowing, potentially carrying the herbicide to other locations.
If possible, it is always better to manage weeds around your palm trees, or to manage unwanted palm seedlings, with a non-chemical control measure. Digging plants out by the roots helps to ensure that they will not return. Exclusion methods like raised beds or containers can keep weeds out of garden areas. Soil solarization, which means spreading black plastic over soil during summer months for a period of several weeks, can kill all weeds underneath, leaving a weed-free bed for planting later on, and is not dangerous when done next to an established tree.