We had hoped to be living in our new house by now, but the builders are still working on it.
(Why don't we use living or just to live there instead?)
The verb to hope uses the particle to when it connects to another verb in the infinitive form:
I hoped to play. (Not "I hoped play")
In other words, it takes an "infinitive complement". That's why you need that to.
Be living uses the so-called continuous or progressive aspect. This aspect is created by using [BE + Present Participle]: [be + living]. So you need that be.
The progressive aspect is used for actions that are ongoing at the time in question. The moment in time mentioned in your sentence is "by now". So, the meaning is "We had hoped that by this moment in time, we would be living in our new house".
Why not "live"? The phrase "by now" invites the use of a "dynamic" verb, a verb that expresses an action. When we use the simple form "live", it denotes a more permanent, long-lasting situation. In other words, the verb "live" acts like a "stative verb":
I hoped to live. (very "stative": I hoped to be in the state of being alive)
I hoped to live in a castle. (less "stative")
I hoped to be living in a castle by January 1, 2015. (more "dynamic")