Rails and Browsersync without a need for Gulp or Grunt. Thanks to @wnstn and @maykefreitas.

bs-config.js

// This gist is based on @wnstn's great [gist](https://gist.github.com/wnstn/36010c8378e850cc3580) 
// and @maykefreitas's [advice](https://github.com/BrowserSync/browser-sync/issues/977#issuecomment-236443965) about turbolinks.

// Code below is generated after running `browser-sync init` in your project's root directory.
// I commented the lines I changed or added so you can run it from scratch as well, if you like.


/*
 |--------------------------------------------------------------------------
 | Browser-sync config file
 |--------------------------------------------------------------------------
 |
 | For up-to-date information about the options:
 |   http://www.browsersync.io/docs/options/
 |
 | There are more options than you see here, these are just the ones that are
 | set internally. See the website for more info.
 |
 |
 */
module.exports = {
    "ui": {
        "port": 3001,
        "weinre": {
            "port": 8080
        }
    },
    // The following was originally:
    // "files": false,
    "files": [
        "app/assets/stylesheets/**/*.scss",
        "app/assets/javascripts/**/*.js",
        "app/views/**/*.erb"
    ],
    "watchOptions": {},
    "server": false,
    // This gets it running properly with Rails
    "proxy": "localhost:3000",
    "port": 3000,
    "middleware": false,
    "serveStatic": [],
    "ghostMode": {
        "clicks": true,
        "scroll": true,
        "forms": {
            "submit": true,
            "inputs": true,
            "toggles": true
        }
    },
    "logLevel": "info",
    "logPrefix": "BS",
    "logConnections": false,
    "logFileChanges": true,
    "logSnippet": true,
    // This snippetOptions bit is new and takes care of turbolinks clashing with browsersync
    "snippetOptions": {
      rule: {
        match: /<\/head>/i,
        fn: function (snippet, match) {
          return snippet + match;
        }
      }
    },
    "rewriteRules": false,
    "open": "local",
    "browser": "default",
    "xip": false,
    "hostnameSuffix": false,
    "reloadOnRestart": false,
    "notify": true,
    "scrollProportionally": true,
    "scrollThrottle": 0,
    "scrollRestoreTechnique": "window.name",
    "scrollElements": [],
    "scrollElementMapping": [],
    "reloadDelay": 0,
    "reloadDebounce": 0,
    "reloadThrottle": 0,
    "plugins": [],
    "injectChanges": true,
    "startPath": null,
    "minify": true,
    "host": null,
    "codeSync": true,
    "timestamps": true,
    "clientEvents": [
        "scroll",
        "scroll:element",
        "input:text",
        "input:toggles",
        "form:submit",
        "form:reset",
        "click"
    ],
    "socket": {
        "socketIoOptions": {
            "log": false
        },
        "socketIoClientConfig": {
            "reconnectionAttempts": 50
        },
        "path": "/browser-sync/socket.io",
        "clientPath": "/browser-sync",
        "namespace": "/browser-sync",
        "clients": {
            "heartbeatTimeout": 5000
        }
    },
    "tagNames": {
        "less": "link",
        "scss": "link",
        "css": "link",
        "jpg": "img",
        "jpeg": "img",
        "png": "img",
        "svg": "img",
        "gif": "img",
        "js": "script"
    }
};

package.json

/*
  *** Don't actually include this comment in the file. Bad JSON. ***
  This file is generated when you `npm init` from the terminal.
  Run `npm install browser-sync --save-dev` to get browser-sync added to `devDependencies` below
  Then manually add the `scripts.browsersync` bit below.
  Finally, open one terminal window for Rails and one for Browsersync.
  Run `rails s` (or `rails -s -b 0.0.0.0:3000` if you have trouble) in one window.
  Run `npm run browsersync` in the other. 
*/
{
  "name": "your-package-name",
  "version": "1.0.0",
  "description": "This README would normally document whatever steps are necessary to get the application up and running.",
  "main": "index.js",
  "directories": {
    "test": "test"
  },
  "scripts": {
    "test": "echo \"Error: no test specified\" && exit 1",
    "browsersync": "browser-sync start -c bs-config.js"
  },
  "author": "@your-handle",
  "devDependencies": {
    "browser-sync": "^2.13.0"
  }
}

Rails error screens are still problematic and require a manual refresh, but oh well.