If a pipe burst in my apartment and I did not have the heat on, am I responsible for all damage it caused?

It would most likely be considered to be negligent to not have the heat one--at least in, say, the mid-50s--in NJ in winter, since the winters easily get cold enough to freeze and burst pipes. If a tenant causes damage to the property through negligence, he or she is liable, or financially responsible, for the resulting damage. From what you write, you would most likely be responsible for the water damage.

If the landlord had not previously repaired water damage from another unit properly, you may be able to 1) argue that some of the damage was from the previous incident and not due to your actions; and/or 2) if  you had to live with damage, you might be entitled to some small amount of compensation for that, which you could apply to offset what you owe. Therefore, you may be able to argue that you are not responsible for the full amount of the costs. You can try to work it out with the landlord; or if you can't and you are sued, raise the above as defenses and/or counterclaims at court.