In the US, do states place restrictions on ownership of firearms out of state? For example, if you legally acquire firearms in one state and keep them in that state when you move to another state, are you required to abide by firearms laws such as registration and restrictions on possession in the state you move to?
The general story is that one state does not have jurisdiction over an act carried out in another state. The Wiki on state gun laws claims that ownership in Illinois requires a permit, but the law is here, and in fact the law addresses possession and acquisition, but not ownership, for example 430 ILCS 65/2:
No person may acquire or possess firearm ammunition within this State without having in his or her possession a Firearm Owner's Identification Card previously issued in his or her name by the Department of State Police under the provisions of this Act
It is legally immaterial that the word "Owner" is in the name of the card. It is likewise claimed that Massachusetts requires a license to own a gun, but from what I can tell the requirements pertain to licenses to possess or purchase, and not just to own. So there seems to be no impediment to actual ownership in the US.
There are lots of different firearms laws and you would have to evaluate them on a one by one basis by looking at its language.
If a state law states that residents of the state with felony records may not own a gun, you might be in violation of that law even if the gun is in a different state. Similarly, if a court order (perhaps a restraining order) insists that you turn over all guns you own to someone, you might be subject to criminal liability or contempt of court liability for not doing so, even if the guns are outside the state. But, a differently worded state law or court order might not have that effect.
A state has no ability to legislatively control what can and cannot be owned in another state. The type of legislation that would affect activities outside of a state's geographic boundaries fall under the concept of "long-arm jurisdiction". States have no jurisdiction on criminal matters, and limited jurisdiction on civil matters, that occur outside its geographic boundaries.
Regarding firearms, specifically, you may find the following helpful.
Your question asks about "ownership" of firearms. Gun control laws generally address acquisition and possession; that is, who can acquire a firearm and who can possess a firearm. Acquisition of the firearm is what comes closest to the word you use: ownership. Acquisition of firearms is not restricted to individuals; trusts and corporations often acquire and possess firearms. It is not at all unusual for an acquirer of firearms to possess or store different firearms in different locations based on desires and localized laws and regulations.
The storage of firearms and firearm components in states other than one's state of "residence" is a common practice. For example, the State of California has many restrictions on the types of firearms that can be possessed within the state and it's not uncommon for people in California to store their California non-compliant firearms in another state such as Arizona. There are businesses such as shooting clubs and ranges that provide storage services for this purpose.
The subject does become particularly complex when explored deeper. As an example, the transfer of firearms can be severely restricted depending on the state and type of firearm or component. Having a friend store a firearm in their home could be considered a transfer and the friend would be considered the possessor of the firearm even if you maintained ownership. Such a transfer or possession could be illegal and subject both parties to criminal prosecution. This is the reason that firearm storage services are hosted by Federal Firearm Licensees (FFLs). FFLs, depending on their license type, have the ability to accept firearms and firearm components.
Don't be confused by state residency, though, as a person can be a resident of more than one state for purposes of acquisition or possession of firearms. It is not uncommon for people to have a residence in one state and a vacation home in another. The Federal Bureau of Alcohol, Tobacco, Firearms and Explosives addresses this issue in an explanation of their regulations:
ATF has previously addressed the eligibility of individuals to acquire firearms who maintain residences in more than one State. Federal regulations at 27 CFR 478.11 (definition of State of Residence), Example 2, clarify that a U.S. citizen with homes in two States may, during the period of time the person actually resides in a particular State, purchase a firearm in that State. See also ATF Publication 5300.4 (2005), Question and Answer B12, page 179. Similarly, in ATF Ruling 80-21 (ATFB 1980-4, 25), ATF held that, during the time college students actually reside in a college dormitory or at an offcampus location, they are considered residents of the State where the on-campus or offcampus housing is located.
In the actual regulations, they give examples of multi-state residents:
Example 2. A maintains a home in State X and a home in State Y. A resides in State X except for weekends or the summer months of the year and in State Y for the weekends or the summer months of the year. During the time that A actually resides in State X, A is a resident of State X, and during the time that A actually resides in State Y, A is a resident of State Y.
The laws that govern the types of firearms and firearm components that can be possessed in a given location are regulated by Federal laws and regulations, State laws and regulations and, in 5 states (Connecticut, Hawaii, Massachusetts, New Jersey and New York), local laws and regulations.