Self-contained poll application
I cannot think of a clean way to extract the validation logic. 


All validation logic should be in an appropriate "validation service" class as @meWantToLearn said.

I assume something like this is possible in FluentValidation:

public class PollVoteValidator : AbstractValidator&lt;Poll,VoteInput&gt; 


If not then make a simple class that contains both of these and then create a validator for that:

public class PollInput {
    public Poll {get; set;}
    public VoteInput {get; set;}
}

public class PollVoteValidator : AbstractValidator&lt;PollInput&gt;


Validation Separation of Concerns

Validations such as if (poll == null) belong in a PollValidator class and if (poll.VoterIps.Contains(voterIp)) in a VoterIpsCollection class.

Even if you want to check for null method arguments up front (and that will simplify downstream control logic), the above still applies. 

Validation Coordination

The PollVoteValidator would coordinate everything perhaps, with logic, somewhere, like this:

public class PollVoteValidator {
    public ValidateResults Validate (poll, voteInput) {
        var pollValidateResults = myPollValidator.Validate(poll);
        var voteInputValidateResults = myVoteInputValidator.Validate(voteInput);
        // I hope poll is not null!
        var voterIpsCollectionResults = myVoterIpsCollectionValidator.Validate(poll.VoterIps);

        if(/* validations above pass */)
            pollVoteValidatorResults = mypollVoteValidator.Validate(poll, voteInput);

        // consolidate validation results as desired (error messages)
        // for return to the caller

        return pollVoteValidatorResults;
    }
} 


One More Thing

This: var voterIp = Request.GetOwinContext().Request.RemoteIpAddress; seems "orphaned" from an OO perspective. voterIp feels like a Voter class property, so maybe you should instantiate a Voter; this implies a "validation service" class for Voter.

The above Voter class stuff may seem like overkill, but there is no OO principle that says "if something seems simple, ignore OO principles".