Yes and no. In principle, the weather doesn't play a major role when exploring a Tyrolean cave, which makes them ideal rainy weather destinations. However, as some can only be reached by hiking, we recommend only visiting these caves in good hiking weather: Kaiser-Max-Grotte, Tischoferhöhle, Obstanser Höhle and Hundsalmhöhle.
In summer in particular, the two ice caves Obstanser Höhle and Hundsalmhöhle are a welcome way to cool off after a hike, which is why we highly recommend them as excursion destinations for hot days.
The Natural Ice Palace and the cave at the Spannaglhaus can be visited in (almost) any weather, as they are only a short walk from the mountain stations of Glacier Bus 2 and 3 on the Hintertux Glacier.