Will I lose anything important if I turn off Google background sync in Android?

If you turn off sync, you won't lose the data that's already uploaded to Google's cloud - contacts, calendar, Gmail messages, and so on will all stay on the servers. You'll only lose the automatic background updates of this information on your phone. So if someone adds an event to your shared calendar or an email arrives, you won't find out about it right away - only when you open the app yourself.

Which apps will actually be affected? Mail will load the next time you open Gmail instead of coming as a notification. Contacts won't sync automatically with the cloud - if you lose your phone, new contacts will stay on the device. Same deal with the calendar. It's not critical if you don't depend on instant notifications and you check the apps yourself every now and then.

The main thing here is that this will noticeably impact your battery life, especially if you have a lot of accounts connected. Just make sure before turning it off that your important data has synced at least once. After that, go to Settings → Accounts → Google, pick your account, and uncheck the services you don't need updating in the background (or turn off sync entirely). You can always turn it back on later if you realize you miss the notifications.

You'll actually lose quite a bit, to be honest. Yeah, your data in the cloud will stay there, but without background sync you won't get real-time updates - emails will come with a delay, your calendar won't refresh until you open the app yourself, contacts you edit on another device won't sync right away. For a lot of people that's a hassle, especially if you're planning your day by calendar or waiting for an important email. Turn it off if your battery's critical or data's expensive, but just keep in mind that you won't have live sync anymore.