Does something that is virtually located on your property like in Pokemon GO incur any kind of property right for the property owner?
In this answer, I address title question: "If someone catches a Pokemon that is on my property, is that theft?"1
The Pokémon is an entry in a database, presented by Nintendo to users in their mobile app and can be included in a user's collections after that user completes some in-game actions. It isn't your property. It isn't even their (the users') property.
Further, a Pokémon appears for all users can be captured again and again by multiple users until it disappears for all users. One user capturing a Pokémon doesn't make it unavailable for others.
The Pokémon doesn't "become subject to" any property rights of the owner of the real property that it happens to be virtually overlaid on. Said another way, it doesn't "incur" any of their property rights. Capturing it in-game is not theft.
1. The question in the question body is "does [a Pokémon] incur any kind of property right for the property owner". Incur means "to become subject to". That is different than the broader verb "implicate". This question does not ask (and this answer doesn't answer) whether the Pokémon can implicate the property owner's property rights (e.g. via attractive nuisance, trespass, etc.). Some of that is addressed at a separate question.
The Pokemon was never in your property! It was displayed on a map as though it was on your property, accessed by feeding the coordinates of your property through the app along with other game data. You do not own your coordinates. The fact that the game encourages people to physically visit the coordinates is an entirely separate issue.
No. In addition to the points made in other answers, you do not have any property rights in the virtual locations of the game. They are a service that is provided to you under a contract with the game operators which allows them to terminate the service at will. The fact that they are mapped by a mobile client so that they appear to coincide with real-world locations is not relevant: you have no rights to specific locations within the game's database, just as you have no rights to tell Google (for example) not to display specific information if somebody clicks on your property's location on Google Maps.
(Since no jurisdiction was specified, I can answer this under Dutch law)
Pokemon Go may be a game by Niantec/Nintendo, but that does not necessarily remove your property rights. The essence of property rights is the exclusive control over anything unique and identifiable. Virtual goods are not exempted, as confirmed in jurisprudence. Gamers can own in-game goods, they can be defrauded out of those goods, and that is potentially even criminal fraud. In particular, the game developer can give property rights to one gamer, to the exclusion of others. Even if the game developer held the right to revoke such rights, this exception only applies to the game developer and not third parties.
Now in this case, there is no exclusive control over the Pokemon. Without exclusive control, there are no property rights, and there cannot be theft.
Exclusive control in this case does not mean sole ownership; there might be multiple owners. To keep in game terms, a clan or guild might have ownership of items that non-members cannot control. However, other items in the world may be first-come first-serve. Those would be an example of items not under exclusive control
This is a new and emerging area of law and there are numerous potential legal theories that could be advanced. All of this is untested.
One thing is clear though, it does not meet the technical definition of theft. Theft requires intending to permanently deprive someone of personal property. That simply doesn't happen here.
However, property rights extend well beyond just the right not to have your property stolen from you. Things that interfere with your enjoyment of your property or affect its value can also give rise to legal actions because they also implicate property rights.
A Pokestop could be an attractive nuisance. It may attract children to your property where hazardous conditions could be present. Generally, a warning is not sufficient to mitigate liability for creating an attractive nuisance. But I don't know if there's even been a case where one party created an attractive nuisance on another person's property without any physical trespass.
Does putting Pokemon on a property virtually change its value? There can be liability for defaming property where that affects its value. Perhaps the people attracted to the Pokemon or Pokestop make the property less desirable.
Numerous other legal theories are possible and all of this is presently untested.
But it's definitely not going to meet the legal definition of "theft".
The other answer here disproves the OPs premise that Pokemon are a physical entity in the world and that they can be property. This answer is instead going to work under the premise and disprove the supposed logic that followed from it.
Let me ask you a question. If someone leaves a box of junk in your backyard because they need somewhere to store, does it automatically become your property? Nope. Even if they no legal permission to put it there, they have 30 days to retrieve their property.
Using this case with the Pokemon, the property does not own the pokemon. Therefore, he cannot claim it was stolen. Technically he cannot give it away either.
Of course, the Pokemon exists for all users so one cannot steal it from another person. So regardless of country, jurisdiction, or rules regarding video game item theft, you cannot be interpreted as stealing the pokemon.