This is a tricky question because the person who caused the burn was not the owner of the property where the accident occurred. Without the third party being involved, the situation would have been a clear-cut claim against the home insurance policy. However, since the burn was caused by someone else, presumably not a dependent of the home owner, the situation is a little more complicated.
The best course of action would be to try to file a claim against the homeowners policy. You'll need to have all of the information regarding your injuries, including the cost of your hospital stay, rehab, and any other procedures which were performed. If the claim can be filed in such a way that you burned yourself rather than received burns caused by someone else, then it would be a simple case of injury liability.
If the claim is denied as a result of the third party's involvement, then your only recourse might be to file a claim or a civil lawsuit against the person who injured you. The person who injured you may have an umbrella policy or other liability insurance that the claim would be accepted by. Without that, legal action is the only way for you to get the injuries paid for.
If the homeowners insurance does accept the claim, then it will go into the owner's CLUE report. Your CLUE report is a collection of insurance related information about you. It includes the number and types of claims that have been filed, as well as other information. When the information about your burn is entered into your friend's CLUE report it can affect their insurance rates for years, especially if this is not the only claim that has been filed.