There is no evidence to suggest the order in which you drink different alcoholic beverages makes a difference in how the alcohol will effect you.
However, people can consume liquor much faster than beer because liquor is more concentrated and therefore a lot less liquid, so they tend to think it's the liquor that got them drunk (or sicker)-but this simply isn't true. You can get just as drunk off of beer as you can liquor if you consume the same number of standard drinks in the same amount of time-your body doesn't process different types of alcohol differently or based on the order you drank them.
One thing to consider is the amount of congeners present in the alcohol you're drinking. Congeners exist in all alcohols, but in different amounts in different drinks. The general rule is darker alcohol=more congeners=worse hangovers. Congeners likely won't impact you while you're drinking, but could make a big difference the morning after. For more info on congeners specifically or hangovers in general, check out the March issue of Husky Headlines which highlights hangovers.
You may have or know someone who may have had an experience that makes you believe that this statement is true. What likely happened is that because the person drank the liquor so much faster than they could drink beer, they got drunk more quickly than they would have had they been drinking beer, but there is no empirical evidence that suggests the order in which someone has drinks makes a difference in the side effects.