Is it true that I don't need to burp my baby if I am nursing?

No. Although many babies burp on their own when nursing, it's still a good idea to burp your baby. You can burp in between breasts or at the end of a feeding.
No. A breastfed baby may or may not burp after a feeding depending on how much air was swallowed during feeding. You should still try though, it may help prevent a big spit up a few minutes later!