Sending connection requests to people whom you don't know at all. The point of LinkedIn is to connect with your contacts. If you try to connect with someone who has no idea who you are, and especially if you don't bother to include a note telling him or her why you'd like to connect, you'll alienate and annoy that person. (And if you send enough of these and in response enough people indicate they don't know you, LinkedIn may even ban you from sending more connection requests.)
Sending connection requests without any context, just the default message. Even with people whom you do know, it's considered good form to personalize the connection request message, even if it's just a line or two. Most folks will still accept the request if they know you, but you'll make a much better impression if you write something personalized to them.
Updating your status too often. LinkedIn isn't Facebook or Twitter; it's a business networking site. If you clog up people's feeds with constant updates or posts that won't be of general interest, you may find some people remove you from their connections entirely.
Contacting strangers about job openings to try to circumvent their company's application system. If an employer has an online job application system, they want you to use it. They don't want you to contact their employees through LinkedIn to ask if they'll pass your résumé along for you. And those employees who don't know you have no reason to vouch for you, after all.
Lying about your title or your job responsibilities. Your co-workers will look at your profile one day, and they will lose all respect for you. And worse, if a reference-checker happens to cross-reference your LinkedIn profile with your résumé and sees discrepancies, that will be a huge red flag.
Indiscriminately endorsing people. Complaints have already started about the abuse of LinkedIn's new endorsement feature, which allows people to endorse you for various skills. You might think you're doing your contacts a favor by endorsing them for a litany of skills, but people don't want their profiles crowded with things they have no real expertise in.
Not building a profile but asking people to connect with you anyway. If you connect with someone and they check out your profile, only to discover that it contains little more than your name, they're going to wonder what you're doing on LinkedIn in the first place. If you're going to use the site, you need to use the site—at least to set up a fleshed-out profile for yourself.
Forgetting that many users can see that you viewed their profile. People with certain types of accounts and certain preference settings can see who has been looking at their profile. There's no need for stealth—their profile is there to be viewed, after all—but if you're looking at it day after day, you'll seem a little stalkerish.
Using groups to try to sell things. LinkedIn's groups are one of the best ways to network and share information. But unless you're in a group specifically designed for selling, you will annoy group members (and maybe even get kicked out) if you try to promote products and services in a space where other people are trying to talk.
Alison Green writes the popular Ask a Manager blog, where she dispenses advice on career, job search, and management issues. She's also the co-author of Managing to Change the World: The Nonprofit Manager's Guide to Getting Results, and former chief of staff of a successful nonprofit organization, where she oversaw day-to-day staff management, hiring, firing, and employee development.