On September 23, 2024, about 1442 eastern daylight time, a Piper PA-28-161, N4298V, was substantially damaged when it was involved in an accident near LaGrange, Georgia. The flight instructor and student pilot sustained minor injuries. The airplane was operated as a Title 14 Code of Federal Regulations Part 91 instructional flight. The flight instructor reported that after completing maneuvers over the local area, the student pilot performed a touch-and-go landing on runway 31 at LaGrange-Calloway Airport (LGC), LaGrange, Georgia. During the initial climb after the touch-and-go landing, about 200 to 300 ft above ground level, the engine began to run rough and the airplane stopped climbing. The flight instructor then took control of the airplane from the student pilot and made a right turn for a forced landing on runway 21. The flight instructor further stated that during the landing attempt, the engine power did not reduce all the way to idle when he pulled the throttle lever fully aft. The airplane subsequently touched down long on the 5,000-ft runway and traveled off the end and down an embankment, coming to rest upright. Examination of the wreckage revealed that the landing gear collapsed and the left wing was substantially damaged. The wreckage was retained for further examination. Additionally, an Avidyne IFD540 was recovered and forwarded to the National Transportation Safety Board (NTSB) Vehicle Recorders Laboratory, Washington, DC, for data download. The engine was further examined at a recovery facility by a representative of the engine manufacturer, under the supervision of an FAA inspector. No issues were identified with the throttle linkage. The examination revealed that the No. 3 cylinder exhaust valve was stuck in the open position; no other anomalies were noted with the airframe or engine. The No. 3 cylinder was then forwarded to the engine manufacturer for metallurgical examination under the supervision of an NTSB investigator. The metallurgical examination revealed rough residues and patches in the valve guide and valve guide material adhered to the exhaust valve stem. Caked residues were observed around the exhaust valve tip. Coked oil deposits appeared to be present in the rocker box around the exhaust valve guide and the cylinder head cooling fins around the exhaust port appeared white around the exhaust port, consistent with heat tinting. Measurements of the exhaust valve guide bore’s inside diameter were all below specification for a new valve guide, and the diametral valve-to-guide clearances were also below specification at the middle and inboard ends of the guide. As of the publication of this report, the engine manufacturer was conducting an internal investigation to address the issue. Download of the Avidyne IFD540 data revealed that no engine data had been stored. Review of maintenance records revealed that the No. 3 cylinder had been operated for about 1,072 hours since it was installed new on December 22, 2021.