Using .htaccess to make all .html pages to run as .php files?

Create a .htaccess file at the root of your website and add this line:

[Apache2 @ Ubuntu/Debian: use this directive]

AddType application/x-httpd-php .html .htm

Or, from comment below:

AddType application/x-httpd-php5 .html .htm

If your are running PHP as CGI (probably not the case), you should write instead:

AddHandler application/x-httpd-php .html .htm 

In My Godaddy Server the following code worked

Options +ExecCGI
AddType application/x-httpd-php .php .html
AddHandler x-httpd-php5 .php .html

This is in edition to all other right answers:

If you are not able to find the correct Handler, Simply create a .php file with the following contents:

<?php echo $_SERVER['REDIRECT_HANDLER']; ?>

and run/open this file in browser.

Output from the php code, copy this output

Use this output in .htaccess file

Create a .htaccess file at the root of your website(usually a folder named public_html or htdocs on linux servers) and add this line:

AddType [[THE OUTPUT FROM ABOVE FILE]] .html .htm

Example

AddType application/x-httpd-php70 .html .htm

Important Note:

If you see blank page or Notice: Undefined index: REDIRECT_HANDLER

Try default in .htaccess

AddHandler application/x-httpd-php .html