Jesse says that he is "Vegan Level 5", which supposedly means he doesn't eat anything that casts a shadow, however everything casts a shadow.
When Lisa sets up camp on the tree branch and comments on the view, we see the Alps, the Eiffel Tower, the Statue of Liberty, the World Trade Center and New York in the distance, all of which being seen from the height of a tree is highly illogical. (Not to mention the geographical inconsistencies.)
The last time when the town of Springfield think a Simpsons family member has died was in Mother Simpson.
Lisa's quest to save Springfield's oldest redwood tree by living in it is a reference to the real-life story of Julia Butterfly Hill and a redwood tree called "Luna". Hill lived in the tree for two years and succeeded in preventing it from being cut down by loggers. The details of Lisa's day-to-day existence in her tree (such as hauling supplies up to her tent by a rope) are very similar to Hill's.
When the giant log destroys Kentucky Fried Panda, Homer exclaims "No! It was finger-ling-ling good!", which is a reference to Kentucky Fried Chicken and its slogan, "Finger-Lickin' Good", and to either Ling-Ling or Ling Ling, both of whom were famous giant pandas.
When Bart tosses his game system in the fireplace, it yells, "No! I can still make you happy! To the max!" The closed captions have the line as, "Warning: store in a cool, dry place. A cool dry place!"
Bart tells Marge that he doesn't have a cap during Marge's "Put it in your cap" song. Apparently he -- and, more realistically, the writers -- forgot the early episodes in which Bart had a lucky red cap.