Are insulin shots suppose to burn?

Yes they do. Yes they do burn, Lantus, (insulin glargine) the long acting insulin tends to burn more so than the shorter acting Apidra, Humalog. Slow injection helps.
Sometimes. Some insulin (glargine) is in an acid solution. Injecting a large dose can sometimes burn. If you use alcohol to clean your skin before injecting, make sure the alcohol is completely dry otherwise it will cause burning. If the insulin has just come out of the refrigerator and is cold, it can feel like burning when injected.