Am I breaking any laws if I burn DVDs to my computer?

Unless you own the material on the DVD or have a license to copy it to your computer, you are probably violating someone's copyright by making an unauthorized copy of the material. There are a few exceptions to this (the material may not be copyrighted, for example) but they're pretty rare.

Likely yes - if the original DVD is protected by DRM (digital rights management) technology. The moment you crack DRM (Digital Rights Management) to rip the DVD, you've violated Title I of the Digital Millennium Copyright Act. 17 U.S.C. 1201 prohibits circumvention of DRM . . . Some courts have tried to leaven this rather harsh rule, but most have not. While it's typically hard to detect small-scale circumvention, the question is whether bypassing DRM is legal. The statute sets up some minor exceptions, but our ripper doesn't fall into any of them. So, the moment a studio protects the DVD with DRM, it gains both a technical and a legal advantage-ripping is almost certainly unlawful.