BE23 Flight Instructor with student reported engine failure during cruise. Flight crew performed an off airport landing.

Narrative

My student and I flew from ZZZ1 to ZZZ in a Beechcraft Sundowner BE23 to get familiarized with the area for his upcoming practical test. A thorough preflight was completed and both tanks were filled and checked prior to leaving. We estimated a fuel burn of 10 gallons per hour for a flight of approximately 1 hour and 15 minutes to ZZZ. Because of the TFR over ZZZ ATC advised us to fly towards ZZZ2 to pick up flight following from ZZZ1 Approach rather than ZZZ Center. We switched tanks every 30 minutes using the Avidyne IFD 440 annunciator a timer on the Transponder; and a cellphone timer. We landed at ZZZ and requested a full stop taxi back to Runway XX. We departed enroute to ZZZ3 for a quick restroom break and climbed to 3500 feet. After reaching our cruise altitude we got the advisory from the Avidyne to switch tanks. My student switched on the fuel boost pump and switched tanks and the RPM dropped; and the fuel pressure dropped to zero. He notified me and I began to troubleshoot the problem while he continued to fly the airplane. I switched back to the right tank and tried recycling the fuel pump while also adjusting the throttle. Power returned intermittently which is when I notified the temporary ZZZ Tower Controller that we were having a problem with fuel and would like to return. We were also looking at possible fields in the vicinity as well. After applying carburetor heat; leaning the mixture and running through the checklist to see if there was any change; we pitched for the best glide speed and requested the closest airport which ATC advised was a private grass strip; ZZZ4. We turned in the direction of the airport using Foreflight. ATC tried to send us in the right direction; but it was extremely difficult to find the grass field amid many grass fields and decreasing altitude. While the power was working intermittently; we descended to 1000 feet and headed toward what we thought was the runway; but once the engine stopped producing power; we committed to the grass field we were aiming for. The aircraft stopped just in front of the tree line with the left wing in the bushes. The airport was just on the other side of the tree line where we made the landing. We secured the airplane; got out to assess the damage and notified ATC by phone that there were no injuries. The time was XA:44. My student did a visual inspection of the tanks; the left tank was at the bottom of the tab approximately 15 gallons and the right wing was approximately 3/4 full and blue staining on top of the right wing. With the help of one of the residents from ZZZ4; we were able to move the plane back to better assess the damage; and discovered that the left landing light lens was cracked. With approval from the FAA we were able to tow the aircraft to the ZZZ4 and tie it down. Total Hobbs time was 2.1 hours.