I think that the recent problems with the Obamacare website will hurt the law. The Obamacare website has been a failure from the start. It is clear that many people will think that the site's problems is going to be a reflection of the problem with the people behind the policy.
Yes, the problems with the Obamacare website will hurt the law, because people no longer believe that the law works. People have given up and have decided just to pay the penalty instead, rather than spend a lot of time on phone trying to sign up. People think the administrators don't know what they're doing.
Opponents of any law will look for any flaw in its implementation to declare the law a bad decision. problems with the Obamacare website not only help fuel the flames of law opposition, but also hurt the law's support from people who would have otherwise supported it. Spending several days trying to access a website doesn't exactly make finding health care a positive experience for anyone.
The problems with the Affordable Care Act website won't hurt the law because laws themselves cannot be hurt. The law has already passed and it will be implemented no matter what Republicans in the House of Representatives say. The website won't hurt the law because Americans need to have health insurance or else pay a fine with their income tax statements every year.
The problems on the Obamacare website were server and developer errors, but has nothing to do with the actual health care law or how effective it will be over the long-run. All indications show that the website has not hurt the law, because enrollment is even higher than they had projected.