Is this a pointer problem?

You are copying the address of the copy of the TreeNode, not the address of the TreeNode in the main.
Pay attention to the commented out function calls and read the following: How to pass objects to functions in C ++?

//Output with TreeNode node as arg
//Creating node with value: 25
//Creating node with value: 15
//Creating node with value: 30
//Creating node with value: 20
//Adding 15 to 25
//15 < 25
//Left node of 25 does not exist
//Adding 30 to 25
//30 > 25
//Right node of 25 does not exist
//Adding 20 to 25
//20 < 25
//Left node of 25 exists 20
//Adding 20 to 20
//20
//20

//Output with TreeNode & node as arg
//Creating node with value: 25
//Creating node with value: 15
//Creating node with value: 30
//Creating node with value: 20
//Adding 15 to 25
//15 < 25
//Left node of 25 does not exist
//Adding 30 to 25
//30 > 25
//Right node of 25 does not exist
//Adding 20 to 25
//20 < 25
//Left node of 25 exists 15
//Adding 20 to 15
//20 > 15
//Right node of 15 does not exist
//15
//30

#include <iostream>
using namespace std;


class TreeNode {
    public:
        TreeNode *left;
        TreeNode *right;
        int value;

        TreeNode(int _value);
        //void add_node(TreeNode  node);
        void add_node(TreeNode & node);
};

TreeNode::TreeNode(int _value) {
    left = 0;
    right = 0;
    value = _value;
    cout << "Creating node with value: " << value << endl;
}

//void TreeNode::add_node(TreeNode node) {
void TreeNode::add_node(TreeNode & node) {
    cout << "Adding " << node.value << " to " << value << endl;
    if (node.value < value) {
        cout << node.value << " < " << value << endl;
        if (left) {
            cout << "Left node of " << value << " exists " << left->value << endl;
            left->add_node(node);
        } else {
            cout << "Left node of " << value << " does not exist" << endl;
            left = &node;
        }
    }
    if (node.value > value) {
        cout << node.value << " > " << value << endl;
        if (right) {
            cout << "Right node of " << value << " exists " << right->value << endl;
            right->add_node(node);
        } else {
            cout << "Right node of " << value << " does not exist" << endl;
            right = &node;
        }
    }
}

int main ()
{
    TreeNode root(25);
    TreeNode n1(15);
    TreeNode n2(30);
    TreeNode n3(20);

    root.add_node(n1);
    root.add_node(n2);
    root.add_node(n3);

    cout << root.left->value << endl;
    cout << root.right->value << endl;

    return 0;
}