Is the possessive "its" or "their" after "all of the country's states and territories"?

Any of these may be correct in a given context. Here are the pronoun antecedents for the two versions and another version that is ambiguous.

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• All of the country's states and territories imposed travel restrictions on its residents.

The antecedent of "it" is "country". Therefore, the states and territories of one country, together, restricted all the citizens of that country.

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• All of the country's states and territories imposed travel restrictions on their residents.

The antecedent of "them" is "states and territories". Therefore, the states and territories of one country restricted all the citizens in their individual domains.

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• All of the countries' states and territories imposed travel restrictions on their residents.

Here, we face a disambiguation problem:

1. If the antecedent of "them" is "country", the states and territories of each country, together, restricted all the citizens of each country.
2. If the antecedent of "them" is "states and territories", the states and territories of each country restricted all the citizens in their own domains.

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Note: I have used "restrict" throughout because it is more convenient than "impose travel restrictions on".

Disclaimer: I am not a native English speaker nor an English teacher, rather an advanced learner. The answer below was written primarily based on common sense and the logic of grammar and might contain minor errors.

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In your sentence, it is the "states and territories" that imposed restrictions and, presumably, each state or territory did so with respect to their respective residents. Therefore, "their" is the preferable option. You can also see it if you replace "all" with "some" in your example:

Some of the country's states and territories imposed travel restrictions on their residents.

— still makes perfect sense.

Some of the country's states and territories imposed travel restrictions on its residents.

— did the states and territories also somehow restrict residents of other areas? On what grounds?

The context and your desired message are also important here. If the restrictions you are talking about are regional, but implemented in every single region, then you would want to choose your second version. If, however, the restrictions are country-level and apply to each region without exception, then I would suggest rephrasing your first sentence, e.g. as follows:

The country imposed travel restrictions on its residents in every state and territory.

— or:

The country imposed travel restrictions on the residents of all its states and territories.