Does the presence of irrational numbers pose any problems for the concepts of limits and continuity?

To the contrary, irrational numbers basically complete the concepts of limits and continuity for rational numbers.
If you take the rational numbers on their own, they are a great many "gaps" between them where the irrational numbers go. (This is rather hard to visualise, because it is both true that there are infinitely many other rational numbers between any two rational numbers and true that you have these gaps.) And one of the ways you can tell these gaps are there are via trying to take limits, because if you just look at the rational numbers there are many sequences that look like they "ought" to converge but don't. (The formal concept for what I'm calling "ought to converge" is Cauchy convergence - roughly speaking, a sequence is Cauchy when terms in the sequence get arbitrarily close together.)
For example: the sequence $3,3.1,3.14,3.141,3.1415,...$ is one composed solely of rational numbers, but it doesn't have a limit in the rationals (since that limit would be $\mathrm{\pi <!--\; \pi \; -->}$). So by working with the notions of limits and convergence you can use the rational numbers to identify the spot in them where $\mathrm{\pi <!--\; \pi \; -->}$ "ought" to be and isn't. And you can actually do this for any irrational number.
So in fact, one of the ways of defining real numbers is by saying "okay, let's extend the rational numbers so that every Cauchy sequence has a limit" (with some compatibility regarding when two Cauchy sequences have the same limit.) That means you add precisely the irrationals. As a result it's actually easier to work with limits and convergence in the reals than the rationals in many ways because unlike in the rationals, every sequence that "looks like it should have a limit" actually does.