If country A and B has no extradition treaty, then can B arrest As citizens residing in B if A requests?
Sure they can arrest them. Are you thinking you can go to a country yours has no extradition treaty with and just commit arbitrary crime sprees with impunity? Presumably you mean to ask if they can extradite someone even without a treaty, and the answer is: depends on their laws, but otherwise of course they can if they feel like it (which may require giving them something in return).
An extradition treaty just creates a formalized process for handling extradition requests, which may include the types of crimes that may be considered, who can be considered or exempted, and the specific procedures a country may go through to decide on whether or not to grant the extradition.
The existence of a treaty does not guarantee the request will be granted. US requests to extradite Roman Polanski from Poland, France, and Switzerland have all failed, despite the fact that the US has extradition treaties with all of them. Several countries, such as France and Canada, will also refuse to extradite anyone who may be subject to the death penalty. This has gotten in the way of US extradition requests before.
I interpret the question a bit differently, as asking can B harass an A citizen in their territotory by arresting them simply because A asked them to? If you are a citizen of A, you can be arrested in B for committing a crime in B. If you are a citizen of A and committed a crime in A (not B) but there is an extradition treaty, you could be arrested and extradited. If there is no crime in B and no possibility of extradition from B to A, then an arrest in B would essentially just be legal harassment. If a nation allows arbitrary arrests, then you could theoretically be arrested without cause. In those nations which require that arrests have some legal basis, you either have to have committed a crime within the jurisdiction, or there has to be a proper extradition request which purports that you committed one of a specified set of crimes in another jurisdiction, in the context of an extradition treaty.