No you can't, you can only see html of any third party website ,there is no tool for checking php code without access to data base.
The best tool to see the HTML is at least IE (both 9 and 10), Firefox, Chrome, Opera and Safari. (You have to check it in all the major browsers.) HTML doesn't have "bugs" (since it's not programming), but errors are best found by experience. "Errors" in HTML are missing tags that are required, improperly closed tags, and mostly the developer not understanding how the document works. (If you don't understand how the document works, the best you can do is throw things on it and hope for the best - which is how most sites appear to have been written.)
I would recommend the built-in function of Google Chrome, you can access it by pressing F12 or right click on the page and select 'Inspect Element' and it will direct you to the line in the HTML file where the element is written. But how can you have bugs in a html file?
Dear irummuneery, You can use http://gtmetrix.com/ it will provide you detail information about improvement your website. Hope it will work for you.