What is this fungus that is growing on my plant, and is it dangerous?
My answer won't apply if your plant isn't a cycad (a species from the genus Cycas) as only cycads are affected. Cycad aren't native to the Dominican Republic but have been planted as ornamentals in many parts of the world. If your plant's a cycad, I'm convinced that what your plant has is NOT a fungal but insect infection - something far more serious for your plant but not harmful for you or other people.

Please compare photos provided to make sure of identification. Even enlarging your top photo,  it's not quite focused or clear enough to be sure. It appears that your white 'fungus' is actually Aulacaspis yasumatsui or the Asian cycad scale insect. Athough native to southeast Asia, it's been spreading to many places where cycad are found. Whether it's reached the Dominican Republic yet, I don't know but its been in Florida since the mid-1990s and is found in a number of Caribbean countries. It's now a very serious pest of cycad worldwide, resulting in the death of many. 

Here are some photos for you to compare. I did my best to find photos that show greater detail for you to check. If the photos fit what you have, you need to treat your plant to prevent it dying and for the infestation spreading to other cycad on the island. I've no experience with this but the information on how to deal with this pest comes from Central Florida Palm &amp; Cycad Society so they should know the best way to treat this. 

Treatment of Asian Scale


 Image from Forestry Images courtesy of Jeffrey W. Lotz 


 Image from Central Florida Palm &amp; Cycad Society