Your obligation is whatever you've agreed upon. Hopefully you have both agreed upon what it is you will be doing for them, in enough detail, that you aren't left wondering if you are expected to fill in content.
Before you take a job, write up a detailed spec list which spells out exactly what you are going to do, and how much you will get paid to do it. The client must sign off on this spec list-- this becomes your contract. Be sure to include how much you will hold hands and how long you will fix bugs for free after the website has gone live. It doesn't hurt to include a paragraph detailing how much you will charge to make any changes after the "warranty period" is over. I also include a paragraph that says "If the client changes these specs, the price will change accordingly."
If your website is not a CMS or database-driven, and your client cannot edit HTML, then obviously you will need to fill in the content, but you should have that in your contract, and you should have that work built into your price.