Father Gilligan in the ballad by Yeats, "The Ballad of Father Gilligan," is tired because his people are dying off so fast.  His job is to go to them, comfort them, and give them last rites when they are near death.  And they are dying so fast that he doesn't get any rest.

Here are the lines that are relevant to your question:

Once, while he nodded on a chair,
At the moth-hour of eve,
Another poor man sent for him,
And he began to grieve.

"I have no rest, nor joy, nor peace,
For people die and die';
And after cried he, "God forgive!
My body spake, not I!'

He knelt, and leaning on the chair
He prayed and fell asleep;
And the moth-hour went from the fields,
And stars began to peep.

Notice, "Another" man sent to him.  Another man is dying.  And the priest responds by beginning to grieve.  The following lines reveal, though, that he is grieving for himself, rather than for the man who is in the process of dying.  He grieves, and he is tired, because he just can't keep up. 

He is so tired that even though he is horrified at what he says--"I have no rest, nor joy, nor peace,/For people die and die"--he falls asleep while praying and leaning on a chair.  

In this poem, I believe that Father Gilligan is tired both in his emotions and as a physical thing.  This is because half of the people in his "flock" are asleep and half of them are dead.  So what does this mean?

The fact that half of them are asleep means that everyone who is alive is asleep.  It must be quite late.  So Gilligan is tired after a long day of trying to care for his people.

The fact that half of them are dead implies to me that Father Gilligan is old and has seen much sorrow.  Because of this, he is emotionally tired.