Are there significant amounts of rare earth elements on Iceland?
The simple answer is no.
There are not any significant rare earth elements (REE) deposits in Iceland. This question: What are rare earths and why do they cluster near alkaline magmatism? gives a list of possible formation mechanisms.
Basically, to form a magmatic deposit of rare earth elements, simple magmatic activity is not enough. It would have to be magmatic activity related to one of two processes (or both):


Magma formed due to very low degree of mantle melting. REE are incompatible elements, meaning that they concentrate into the liquid during melting. If the mantle melts to a very low degree (let's say less than 1%), REEs are concentrated in it. Once the melting proceeds, the REE-rich liquid is diluted by further REE-poor melt. Rocks that form in this process are, or example, carbonatites and alkaline magmas. They do not exist in Iceland (and if they do, very scarce).
Magma that has evolved and differentiated. Let's say we take the REE-poor magma from 1. Now it starts crystallising. Remember that REE are incompatible and will stay in the liquid. So the residual melt eventually gets enriched in the REE. Think salt water being evaporated - the salt remains inside the water making it more saline. Pegmatite is one type of rock that forms in this process. Similarly, it does not occur in Iceland.