Dublin mass transit does not allow you to use a contactless debit/credit card or mobile payment directly when boarding, but the ticket machines do accept them.
The Leap card is the contactless transit payment card for transit in Dublin (and various other major Irish cities). They are available from convenience stores, and can be refilled at convenience stores and at DART/Luas ticket machines. There are no ticket machines for buses; other than the airport bus, you must use either a Leap card or coins (bills are not accepted).
Virtually all restaurants, stores, etc. accept contactless cards and mobile payment.
Some hotels, touristy gift shops, etc. might accept dollar bills but it wouldn’t be common. The national currency is the euro. (Cards denominated in other currencies are still accepted, of course.)