Yikes, this is tight.
If this a single ticket starting, United is responsible for getting you to your final destination if you miss a connection without being at fault. So if you miss connection United will rebook you to your final destination but you may end up on a different routing and not on a Lufthansa flight.
Chances are, if you miss it, they will just put you on the same flights the next day, so you need to decide whether that would be an acceptable outcome. They may give you hotel and food vouchers, they may not (depends on a lot of complicated factors and is partially in the discretion of the gate agent).
The flight is within the legal connection time for IAD, so you have an ok chance of making it, if everything goes well. Your bags will be checked through to your final airport. It's basically a domestic connection: you get off your plane, run like heck to the gate for your departure and hope you get there in time to still get on. Since it's an international departure, there may be an extra doc check along the way. There is really no room for error and delays.
The actual chance depends on your exact itinerary: I would check what gates/terminals the flights typically arrive and depart and what the on-time record of the incoming flight is. If you have to ride the train from Terminal A to Terminal C or D, you are unlikely to make this.