If that plant is a banana:
Do not do any pruning back the leaves until your risk of frost is completely over. These dead leaves will help protect/mulch the banana from frost. I usually wait until I see some green growth coming out from the banana and if there is no more frost coming then I go ahead and cut off the dead stuff.
In general with all plants, you should leave the pruning till spring is certainly there.
As with palm trees it will depend on what kind of palm tree it is. Some are more cold hardy than others. Usually if the palm crown freezes totally and the spears pull out then it is usually dead. You should post more detailed photos of your palm tree with name of palm on the palm tree forum and they will be able to tell you more information.