Wary is the feeling of danger and be cautious about it. It entails carefully watching and guarding against deception.
"Be wary of the dog, it is unpredictable."
Weary is an expression of fatigue though it could accommodate mental tiredness too.
"The thought of paperwork I have to do is weary enough."
Leery is a state of being cautious, hesitant or nervous about something. It entails having reservations and concerns for an action.
"Stop being leery, every thing will be alright."