Actually you can run node.js on a typical shared hosting with Linux, Apache and PHP. Even NPM, Express and Grunt work fine. Here are the necessary steps:

1) Create a new PHP file on the server with the following contents and run it:

<?php
//Download and extract the latest node
exec('curl http://nodejs.org/dist/latest/node-v0.10.33-linux-x86.tar.gz | tar xz');
//Rename the folder for simplicity
exec('mv node-v0.10.33-linux-x86 node');

2) Install your node app, e.g. jt-js-sample, using npm:

<?php
exec('node/bin/npm install jt-js-sample');

3) Run the node app from PHP:

<?php
//Choose JS file to run
$file = 'node_modules/jt-js-sample/index.js';
//Spawn node server in the background and return its pid
$pid = exec('PORT=81 node/bin/node ' . $file . ' >/dev/null 2>&1 & echo $!');
//Wait for node to start up
usleep(500000);
//Connect to node server using cURL
$curl = curl_init('http://127.0.0.1:81/');
curl_setopt($curl, CURLOPT_HEADER, 1);
curl_setopt($curl, CURLOPT_RETURNTRANSFER, 1);
//Get the full response
$resp = curl_exec($curl);
if($resp === false) {
    //If couldn't connect, try increasing usleep
    echo 'Error: ' . curl_error($curl);
} else {
    //Split response headers and body
    list($head, $body) = explode("rnrn", $resp, 2);
    $headarr = explode("n", $head);
    //Print headers
    foreach($headarr as $headval) {
        header($headval);
    }
    //Print body
    echo $body;
}
//Close connection
curl_close($curl);
//Close node server
exec('kill ' . $pid);

Voila! Have a look at the working demo of a node app on PHP shared hosting.

EDIT: Here is my new project Node.php on GitHub.