They do. The acid in lemon juice and vinegar can trigger the same chemical reactions as sulfuric acid, but they're much weaker and the reactions aren't nearly as intense. Lemon juice can be used to make a battery, can cause mild burns on sensitive parts of your skin, and can even "cook" food in ways very similar to weak sulfuric acid.
The sour taste of lemons and vinegar is a sign of the acid in them. I accidentally got a taste of dilute sulfuric acid once: it tasted like the sourest lemon you could possibly imagine. (Do not try this at home, obv.)
Acids (and their chemical antitheses, "bases") have their strength measured in a scale called pH (that's labelled correctly). pH stands for "potential of Hydrogen", which means how aggressively the acid will work to take or acquire a hydrogen atom from a substance it comes in contact with, or how likely to donate one, in the case of an alkaline solution. (Alkaline is the technical name for a "base".)
The pH scale reads from 14 (for the strongest bases, such as chlorine bleach) to 0 (for the strongest acids).
In general, as others have noted, citric acid (lemon juice) is less aggressively acidic than battery acid, sulfuric or hydrochloric acid, to name some of the more potent ones that we may encounter. (Most of the common things that we drink are mildly acidic, including milk and orange juice. Plain tap water generally has a "slightly acid" reading - just below 7 on the pH scale, and your blood has a "slightly alkaline" reading, just above 7. A pH of 7 is absolute neutrality, neither acid nor alkaline. Lemon juice and vinegar are generally around pH 2 or a little higher - still plenty acidic, but not as strong as the strongest acids.)
Have a look at the Wikipedia entry for pH for more understanding and a visual representation of some common pH values: https://en.wikipedia.org/wiki/PH
They all burn, its called an exothermic chemical reaction, meaning that it reacts with things that it comes into contact with that are capable of reacting. This means it alters physical objects it comes into contact, and produces heat. The degree is based on how concentrated the acid is, and how strong the acid itself is.
Lemon juice is a mild acid and is very dilute, meaning that you can touch it with your skin or even ingest small quantities without sustaining damage, however with sustained contact it may lighten skin, hair, and damage your teeth. When it reacts it is also producing tiny amounts of heat, but probably not enough to notice or offset the cooling effects of the water.