Would any fire based attacks be "effective" against any wood object, such as a tree or a wooden house, or is that just fully up to the GM?
Also, if you catch an item on fire, would the burn damage bypass Hardness?
Energy Attacks: Energy attacks deal half damage to most objects. Divide the damage by 2 before applying the object's hardness. Some energy types might be particularly effective against certain objects, subject to GM discretion. For example, fire might do full damage against parchment, cloth, and other objects that burn easily. Sonic might do full damage against glass and crystal objects.
Bolding mine. In other words, it's up to the DM to apply some common sense. A sudden blast from Burning Hands isn't going to light anything on fire, but deliberate attempts at trying to burn something down shouldn't be hindered.
That said, an actual tree is pretty hard to set aflame. Dried branches and logs are one thing, and even they need minutes (dozens of rounds) to get going in a campfire. I see no issue with a tree benefitting from the half damage and/or hardness rules, unless there are particularly dry conditions that could lead to a forest fire. A single bomb against a normal healthy tree will likely just leave a scorch mark.