Let's live in hope. Let's live on hope. Let's live with hope.
Which preposition should be put ?
There is a difference in meaning between prepositions.
To live in hope = Hope surrounds us like a house, we feel comfort that comes from it.
To live on hope = Hope is like the fuel or food for us, it sustains us.
To live with hope = Hope lives alongside us, it is part of our life, like a dear family member.
So, it depends on how you want your statement to be interpreted.