I have found myself weary lately in spirit especially, but somewhat in body as well. What keeps me going is the faint echo I hear of hope, and then God's encouraging to my spirit and mind along the way.
Weariness can be symptomatic of sin, or maybe simply of physical fatigue. Sin can be as subtle as not drawing near to God, not seeking God wholeheartedly. Though I find God's forgiveness and sense of presence can come just as quickly. The problem comes when I encounter this weariness over a few days. One can get to the place where it seems like one is going to give up and quit.
What is difficult about being weary is that any better sense is honestly beyond one's self. But for us in Jesus, we know that we can't think that when we run out of vigor and life, then that's the end. If that's the end, then we're not getting what we need from God. And from God in Christ we get all that we need for a godly life (2 Peter 1).
Weariness can help us unlearn all of what in us is not pleasing to God. Weariness can be a pointer that in some way there is something displeasing to God about us. And as God's children we don't find it pleasing either, since God does not stop his good work in us, and works in us both to choose and to do his good will.
But weariness is a dangerous time. What if we give up, and in so doing give in to that which might be damaging to us, as well as to others (Psalm 73)? What if we begin to go a completely different way, the way of the world- as did Demas? Or at least commit a sin that wreaks havoc in our world? We best beware and not go there. Better to take Jesus' words seriously and radically deal with any such thoughts.
If weary, wait. Let's wait on God; let's seek God and renewing of communion with him- and from that with each other in Jesus. And let's keep doing it.