Why do mutations not take place in mRNA of higher eukaryotes?
The premise of the questions suggests that mutations cannot take place in the mRNAs of higher eukaryotes. 
To answer your question I think it is important to consider two viewpoints:

First, from a theoretical point of view, since DNA and RNA are as you pointed out composed of nucleic acids, they both can be mutated if enough energy is provided (UV light, chemicals, etc) which invalidates the premise of the questions.

Now, from a practical point of view, as you mentioned most mRNAs molecules have a short half life typically in the minute to day range whereas DNA molecules exist during the whole existence of the organism.

While it can occurs that mRNAs are mutated people are not interested to study this aspect for the following reasons:

You pointed out in the comment section, a mutation in a mRNA molecule might lead to translated malformed protein which can easily be degraded. It will be only one mRNA transcript from thousands transcripts. With a short half life, the mRNA an proteins will be degraded which will not have a long standing impact of the cell/organism. As such it will be very difficult to observe a phenotype which will affect the whole host.

Thus RNAs mutations have only transient effects which will not affect the host in the long term. RNAs molecules are not more protected than DNA molecules just that they are short lived so the host is protected from the effects of RNA mutations.

Hope this helps!