The building that housed the school was run down, with paint peeling off the walls and all the furniture like chairs and tables falling apart.

peeling 是动词 peel 的现在分词形式。peel 本意是"剥落、脱落"，指外层物质（如油漆、皮肤等）从表面分离脱落的过程。peeling off 在这里描述油漆从墙面逐渐剥落的状态。