Creating and developing own tumblr themes has always been a pitty. However, this single line of javascript will help you A LOT. Follow the instuctions in the first comment below to get your own live version of your theme.

source.js

setInterval(function() {
    jQuery.ajax('YOUR-URL-TO-THE-THEME-FILE', {cache: false}).success(function(html) {
        var btn = jQuery("div[data-action='update_preview']").first();
        if (html!=ace.edit('editor').getValue()) {
            ace.edit('editor').setValue(html);
            if (!btn.hasClass('disabled'))
                btn.click()
                
        }
    });
},1000);

Update, edit and develop Tumblr Theme dynamically.js

setInterval(function(){jQuery.ajax('YOUR-URL-TO-THE-THEME-FILE',{cache:false}).success(function(html){var btn=jQuery("div[data-action='update_preview']").first();if(html!=ace.edit('editor').getValue()){ace.edit('editor').setValue(html);if(!btn.hasClass('disabled'))btn.click()}});},1000);

I got some security problems using this snippet.
As the tumblr customize page is on https and my static files are served over http, when I run the script from the console i get this javascript error:

Mixed Content: The page at 'https://www.tumblr.com/customize/BLOG_NAME' was loaded over HTTPS, but requested an insecure XMLHttpRequest endpoint 'http://MY_FILE_URL?_=1432649213872'. This request has been blocked; the content must be served over HTTPS.

Any idea how could I solve that?

In chrome, you can allow mixed content by clicking on the shield in the right side of the address bar and selecting "allow insecure scripts to run"
Hope this helps!