That is a Sarracenia pitcher plant. They're from the bogs of North America, and will almost certainly not survive in a vivarium. They need a dormant period that simulates winter conditions. For vivariums, you need tropical pitcher plants of the genus Nepenthes. You can tell them apart by the leaves. Nepenthes are usually vining plants that have pitchers at the end of a petiole that is broad and leaf shaped. Those that don't creep grow in bunches, but you can usually see the broad petioles, though they are usually buried in some species.
In answer to your question, no, most darts are not harmed by pitcher plants. For Nepenthes that are small enough to be in a viv, the non-thumbnail frogs are either too big to fit in or are more than capable of climbing out.