Trespassers and even burglars are not necessarily denied recourse for a dog bite. Under rare circumstances, even a burglar can recover compensation. The analysis is complicated.
All dog bite statutes exclude burglars and other trespassers. Therefore the only way a burglar can recover compensation would be under the common law (i.e., stemming from English law and modified by court cases in the state having jurisdiction). The common law says that when a person has an animal known to cause injury because of a dangerous propensity, there is strict liability. This is the so-called "one-bite rule" but it applies to all domestic animals and all types of injuries.
Under the old common law, burglars could be compensated because the policy of the law was dead-set against having such animals. However, modern cases confirming the common law position are rare. One reason is that most dog bite cases are brought under dog bite statutes; another is that tort reform and traditional negligence rules place blame on wrongdoing victims and cut their recovery even when they have a viable cause of action. In other words, if a burglar gets bitten by a dog, a jury might say that the burglar wins if the dog was known to be a killer of people, but the jury might also say that the incident was 95% the fault of the burglar for being there in the first place. In that event, his $50,000 verdict would produce an award of $2,500 in a strict comparative negligence state, but NOTHING in a modified comparative negligence state.
Another complication in this analysis has to do with the doctrine of trespass. The dog bite statutes and other state and federal law give a range of people the right to go onto the property of others. Examples are police, firemen, postal workers and utility workers. Additionally, there are many people whom a landowner is "deemed" to have invited onto the property. The law uses the term "implied invitation" for visitors to a garage sale, real estate agents who have a listing, delivery people, and many others. There also are those who are expressly invited, like schoolmates and other guests. While a burglar is, by definition, not an invitee, all of these other people are. Therefore, they are not considered to be trespassers.
As a practical matter, attorneys do not represent victims in cases where the likely recovery will not be sufficient to pay a fair amount to the victim, pay or reimburse the medical treatment costs (past and future), compensate the attorney for his or her services, and reimburse the attorney or the client for the costs of the case. Therefore a burglar or trespasser with a $100,000 injury might not find a lawyer willing to represent him.