How to verify user's current password when changing password on Firebase 3?

You will be able to achieve it using reauthenticate before changing the password.

let user = FIRAuth.auth()?.currentUser
let credential = FIREmailPasswordAuthProvider.credentialWithEmail(email, password: currentPassword)    

user?.reauthenticateWithCredential(credential, completion: { (error) in
    if error != nil{
        self.displayAlertMessage("Error reauthenticating user")
    }else{
        //change to new password
    }
})

Just to add more information, here you can find how to set the credential object for whatever provider you are using.

For Swift 4:

typealias Completion = (Error?) -> Void 

func changePassword(email: String, currentPassword: String, newPassword: String, completion: @escaping Completion) {
    let credential = EmailAuthProvider.credential(withEmail: email, password: currentPassword)
    Auth.auth().currentUser?.reauthenticate(with: credential, completion: { (error) in
        if error == nil {
            currentUser.updatePassword(to: newPassword) { (errror) in
                completion(errror)
            }
        } else {
            completion(error)
        }
    })
}

Firebase Documentation can be found here

For Swift 5 to change password in firebase

import Firebase

func changePassword(email: String, currentPassword: String, newPassword: String, completion: @escaping (Error?) -> Void) {
        let credential = EmailAuthProvider.credential(withEmail: email, password: currentPassword)
        Auth.auth().currentUser?.reauthenticate(with: credential, completion: { (result, error) in
            if let error = error {
                completion(error)
            }
            else {
                Auth.auth().currentUser?.updatePassword(to: newPassword, completion: { (error) in
                    completion(error)
                })
            }
        })
    }

How to use ?

self.changePassword(email: "[email protected]", currentPassword: "123456", newPassword: "1234567") { (error) in
        if error != nil {
            self.showAlert(title: "Error" , message: error?.localizedDescription)
        }
        else {
            self.showAlert(title: "Success" , message: "Password change successfully.")
        }
    }