Most reasonable companies that hire you will be willing to wait for you to start until you've "served" out your remaining time at your current job. If they're demanding you start yesterday, you might consider why they're so desperate to have you, and whether you really want to work for such a company.
Why You Should Give Notice
While there are no legal implications to not giving notice, it is courtesy, and not just to your boss. Consider the people you work with and any others who might be burdened with your workload if you cut out early. Add to that the fact that they will be scrambling to find someone to replace you, and you've instantly created enemies by leaving early.
And while you may not be able to imagine wanting to come back to work for this company, you never know. It's always wise to stay in your boss's good graces in the event you need a reference letter or recommendation down the road.
Also, depending on what your human resources policy on unused vacation time is, you might lose that time you have accrued if you don't provide sufficient notice before leaving. Check your company handbook to see what the rules are on this one. It's probably not worth it to forsake your hard-earned vacation time just to avoid working 10 more days.
When You Don't Have To
Some bosses will simply let you go once you put in your notice. Whether that's due to a temper tantrum, or just your boss's style, don't worry about it ... unless that means you'll have two weeks where you're not earning income. If you know your boss has always fired every employee who ever put in his resignation, you might take your chances and quit just a few days before you're due at the new company to keep the cash flow steadier.
If you feel you have been harassed or verbally abused, there's no benefit to staying. If you've done your best to rectify the situation, you're probably not too worried about getting a reference from this job anyway. Take your sanity and go. Likewise, if your current job has caused undue stress, ask yourself what the benefits to your health are, if any, in staying. Likely none.
No Easy Answer
Each situation is different when it comes to quitting and giving notice. Assess what the benefits will be if you stay a few more weeks: a recommendation, no burned bridges, co-workers who won't give you the evil eye if they run into you. Then look at what you might gain if you leave early. Decide which provides more benefit to you, and do what you feel is right.
Always be open in your communication. If you feel like you need to cut out before your two weeks, explain to your boss or HR manager why. You might find her to be sympathetic if your reason for leaving is genuine. This might help smooth over any ruffled feathers if they find your departure abrupt.
Lindsay Olson is a founding partner and public relations recruiter with Paradigm Staffing and Hoojobs.com, a niche job board for public relations, communications, and social media jobs. She blogs at LindsayOlson.com, where she discusses recruiting and job search issues.