How can I let players "scrape away" one texture to reveal another?

If you're okay with mesh colliders, take a look at this. The example describes a (primitive) version of what you want.

The next step would probably be to write a shader that takes three textures: a "rusty" version, a "clean" texture and a "splatmap" that stores where to apply which texture. Instead of the main texture, the script should paint into this splatmap.

All the shader needs to do is "fade" between the two textures. I'm at work right now and can't give you all the boilerplate from memory, but the relevant line should look something like this:

out.rgba = _rusty.rgba * _splat.r + _clean.rgba * (1.0 - _splat.r)

I'm using the splatmap's red channel here, any other will do as well. The splatmap can have a different resolution than your other textures so play around with that and pick whatever looks best on your object.

EDIT: Here's the final code that did the trick (made by OP). DotClick.cs should be applied to the camera and ClearableShader should be applied to the object in question.

ClearableShader.shader:

Shader "Custom/ClearableShader"
{
    Properties
    {
        _Color ("Color", Color) = (1,1,1,1)
        _CleanTex("Clean Texture", 2D) = "white" {}
        _RustTex("Rust Texture", 2D) = "white"
        _SplatTex ("Splat Map", 2D) = "white"
    }
    SubShader
    {
        Tags { "RenderType"="Opaque" }
        LOD 200

        CGPROGRAM
        // Physically based Standard lighting model, and enable shadows on all light types
        #pragma surface surf Standard fullforwardshadows

        // Use shader model 3.0 target, to get nicer looking lighting
        #pragma target 3.0

        sampler2D _CleanTex;
        sampler2D _RustTex;
        sampler2D _SplatTex;

        struct Input
        {
            float2 uv_CleanTex;
        };

        half _Glossiness;
        half _Metallic;
        fixed4 _Color;

        // Add instancing support for this shader. You need to check 'Enable Instancing' on materials that use the shader.
        // See https://docs.unity3d.com/Manual/GPUInstancing.html for more information about instancing.
        // #pragma instancing_options assumeuniformscaling
        UNITY_INSTANCING_BUFFER_START(Props)
            // put more per-instance properties here
        UNITY_INSTANCING_BUFFER_END(Props)

        void surf (Input IN, inout SurfaceOutputStandard o)
        {
            fixed4 rustTex = tex2D(_CleanTex, IN.uv_CleanTex);
            fixed4 cleanTex = tex2D(_RustTex, IN.uv_CleanTex);
            fixed4 splatTex = tex2D(_SplatTex, IN.uv_CleanTex);

            o.Albedo = (rustTex.rgba * splatTex.r) + (cleanTex.rgba * (1.0 - splatTex.r));
        }
        ENDCG
    }
    FallBack "Diffuse"
}

DotClick.cs

// Write black pixels onto the GameObject that is located
// by the script. The script is attached to the camera.
// Determine where the collider hits and modify the texture at that point.
//
// Note that the MeshCollider on the GameObject must have Convex turned off. This allows
// concave GameObjects to be included in collision in this example.
//
// Also to allow the texture to be updated by mouse button clicks it must have the Read/Write
// Enabled option set to true in its Advanced import settings.

using UnityEngine;
using System.Collections;

public class DotClick : MonoBehaviour
{
    public Camera cam;

    void Start()
    {
        cam = GetComponent<Camera>();
    }

    void Update()
    {
        if (!Input.GetMouseButton(0))
            return;

        RaycastHit hit;
        if (!Physics.Raycast(cam.ScreenPointToRay(Input.mousePosition), out hit))
            return;

        Renderer rend = hit.transform.GetComponent<Renderer>();
        MeshCollider meshCollider = hit.collider as MeshCollider;

        if (meshCollider == null|| rend == null || rend.sharedMaterial == null || rend.material.GetTexture("_SplatTex") == null)
            return;

        Texture2D tex = rend.material.GetTexture("_SplatTex") as Texture2D;
        Vector2 pixelUV = hit.textureCoord;
        pixelUV.x *= tex.width;
        pixelUV.y *= tex.height;

        Circle(tex, (int)pixelUV.x, (int)pixelUV.y, 10, Color.red);
        // tex.SetPixel((int)pixelUV.x, (int)pixelUV.y, Color.red);
        tex.Apply();
    }

    // Taken from https://answers.unity.com/questions/590469/drawing-a-solid-circle-onto-texture.html
    public void Circle(Texture2D tex, int cx, int cy, int radius, Color color)
    {
        int x, y, px, nx, py, ny, d;

        for (x = 0; x <= radius; x++)
        {
            d = (int)Mathf.Ceil(Mathf.Sqrt(radius * radius - x * x));

            for (y = 0; y <= d; y++)
            {
                px = cx + x;
                nx = cx - x;
                py = cy + y;
                ny = cy - y;

                tex.SetPixel(px, py, color);
                tex.SetPixel(nx, py, color);

                tex.SetPixel(px, ny, color);
                tex.SetPixel(nx, ny, color);
            }
        }
    }
}