Carry a boolean value from scene to scene?

If you want just few boolean values then I'd recommend just to set them as static variables, like this:

 public static bool weapon = true;

Though if you have plenty of them and want to keep them with whole class/object holder or you might happen to have few instances of that class and want to keep values independant you might want to use DontDestroyOnLoad.

Just read through both links that I've provided above^ and choose whichever suits you more.

Add the static keyword:

   public static bool weapon;

Another option is to have some type of Manager behaviour that has DonDestroyOnLoad called.

I recommend you check out PlayerPrefs. "PlayerPrefs".

Here is you're script modified that should work. Also with this method you could add a "save" feature. Now since you did not provide me with a full code you will need to make modifcations but you should get a base idea. (If you want sent the full code in a comment and I will add this stuff)

 using UnityEngine;
 using System.Collections;
 
 public class KeepVar : MonoBehaviour {
 
     public bool weapon;
     int weaponInt = 0;
     public string Scene;
 
     // Use this for initialization
     void Start () {
         if (PlayerPrefs.GetInt ("pweapon") == 1) {
             weapon = true;
         } else {
             weapon = false;
         }
     }
     
     // Update is called once per frame
     void Update () {
         if (weapon == true) {
             weaponInt = 1;
         }
         if (weapon == false) {
             weaponInt = 0;
         }
 
         if (Input.GetKeyDown (KeyCode.W)) {
             PlayerPrefs.SetInt("pweapon", weaponInt);
             Application.LoadLevel(Scene);
         }
     }
 }