Mournful means sad and sorrowful, and numbers means tunes or songs. The speaker here asks others not to sing sad and pessimistic songs about life’s uselessness when he says “Tell me not in mournful numbers, / Life is but an empty dream”. The speaker is an optimistic young man who disapproves of the pessimistic theory of life and believes that this life is real and important.

So, “mournful numbers” here refers to the complains and whines of unsuccessful people who think that nothing can be achieved in this life.