It depends on which Rite your church follows.
Ash Wednesday is not a specifically Roman Catholic peculiarity. Ash Wednesday is the beginning of Lent in the Western Rite of Christianity, and, therefore, it is practised in the Latin Rite of the Roman Catholic Church, in many Anglican, Lutheran, Methodist &c. churches, and also in Orthodox churches and monasteries that use the Western Rite.
However, in the Eastern Rite, the beginning of Lent is not marked with an Ash day or ceremony as far as I am aware.
So regarding your question, yes Ash Wednesday does exist in Orthodoxy and yes you should be burning your palm frond but only if your church follows the Western Rite. If you're Eastern Rite, then the answer is no to both questions.
I hope this helps.