The uncertainty principle is about unavoidable measurement "errors" (measurements that can’t be done). If you are going to measure the time (or anything) you will always get finite values. Finite values are not irrational.

Either one talks about the underlying physics or about the measurements of it. If one talks about the underlying physics all bets are of, it is unmeasurable by definition. Anything is possible bellow the measurement threshold(including irrational numbers).

If one talks about the measurements you will always get finite rational values.