Applications of uncountability of the real numbers

It gives a short argument for the fact that the vector space dimension of $\Bbb R$ over $\Bbb Q$ is infinite: $$ \dim_{\Bbb Q}(\Bbb R)=\infty. $$

Also, there is an application for computer science. Consider the set of functions that take an integer argument and return an integer result. This set is uncountable.
Since the set of computer programs is countable, there are uncomputable functions.

In analysis and topology it is common to study sets of reals by classifying them according to some notion of "size". This is done when studying Lebesgue measure, or Baire category, or many other ("ideal-based") notions. Typically, in any of these contexts, we analyze sets of reals by discarding a "small" fragment and focusing on the rest. None of these notions would survive without the reals being uncountable. For instance, the countable union of singletons has measure 0, so any set of reals would have measure zero. This would make it impossible to develop the very useful integration theory of Lebesgue. Baire category is commonly used to establish existence results in settings where explicit construction of the desired objects is cumbersome or not feasible. Again, the countability of the reals would remove this approach from our toolbox.