Several effects may happen, depending on the size of the mountain and which desert it pops into.
If the mountain is high enough to block clouds, it will cause rains on the side that does the blocking. That may create an oasis by the mountain base. If those clouds would regularly go rain somewhere else, you may turn that somewhere else into a desert.
Also notice that the wind may pick up sand from deserts and deposits that sand continents away. That sand may be important to another ecosystem. From NASA:
NASA Satellite Reveals How Much Saharan Dust Feeds Amazon’s Plants
A single mountain might not block the Sahara, but a mountain range might.
Last but not least. Sometimes a desert area exists within mountain ranges (i.e.: a region of the Sahara called the Tibesti, not to be confused with the Tibet, though Tibet is also desertic around some peaks). Dropping a mountain on a mountain may cause
Xzibit to exclaim "yo dawg" a landslide of biblical proportions, potentially causing a big earthquake in the region. At the very least the original mountain would never be the same.