There are hazards associated with any chemical reaction, though some receive more attention than others. Danger is clearly associated with fires and explosions, which are produced by exothermic reactions that can release tremendous amounts of energy. Still, even endothermic reactions, which take in energy, have the potential to cause harm. Proper safety measures must be taken, as is always the case in chemistry.
Personal Protective Equipment (PPE)
Good lab practices include wearing proper protection. A lab coat or apron is always a good idea to protect skin and clothing from unexpected splashes or spills. Similarly, goggles are wise to protect the eyes. (Note - goggles worn on the forehead are not protecting the eyes!) Gloves are often recommended to protect hands from chemical exposure.
In the case of endothermic reactions, which may result in substantially lowered temperatures, gloves may also be desirable to protect against frostbite. While latex, nitrile, or rubber gloves are suitable for protection against chemicals, simple cloth gloves work well against cold. If both forms of protection are needed, gloves can be layered. Two pair of latex gloves provide better thermal insulation than a single layer, or simply slip a pair of cloth gloves over the rubber gloves.
It is standard practice to check for cracks, stars, and deep scratches in glassware before heating. The same should hold true before performing any reaction that results in a temperature drop. Glass contracts when cooled, and can cause cracks to spread, breaking or shattering the flask and spilling its contents. (Think about what happens to small cracks in windshields in the winter.)
If the reaction must take place in a stoppered flask, it is important to consider your choice of stoppers. Unless a gas is generated by the reaction, a temperature drop also means a pressure drop inside the flask. This will work to draw the stopper into the flask, and can make it difficult to remove safely. Glass stoppers, which are rigid and expand/contract at the same rate as the flask itself, are a good choice. They may initially seem stuck, but gentle heating of the neck of the flask will usually release them. A gentle tapping on the side of the stopper can also help to release it. Never rely on the application of brute force, as you may break the neck of the flask rather than removing the stopper. Cork, rubber, and Teflon stoppers can all be compressed, and will be sucked more deeply into the flask. Upon warming, they expand, and become wedged more tightly within the neck of the flask. Patiently working the stoppers back and forth may release them. In the worst cases, these stoppers can be removed by drilling them or with a cork remover (as for wine bottles). Again, excessive force is unwise. If breaking the flask is the only option, consider using a glass cutter on the neck of the flask, so that you can achieve a clean break and control the number of pieces. Always dispose of the broken glass in an appropriate container for sharps or broken glass.
Every chemical is hazardous unto itself. Make sure to know the properties of both the reactants and the products before beginning any experiment. Consult an MSDS sheet if any of the chemicals are unfamiliar to you, so that you will know safe handling and disposal requirements.
Consider the difference between two common endothermic reactions. In the first case, solutions of citric acid and baking soda (sodium bicarbonate) are mixed. Both compounds are fairly innocuous, with little risk beyond mild irritation if inhaled or exposed to the skin. Similarly, their products are not terribly threatening, and the whole mixture can be disposed of down the drain. A second case is the reaction between the solids barium hydroxide octahydrate and ammonium thiocyanate. These reactants are more toxic to begin with, and potentially worse after reaction. The reaction produces ammonia gas - potent enough to cause a painful experience if inhaled or exposed to the eyes, where it forms ammonium hydroxide - a strong base. Barium itself is a toxic heavy metal, so barium compounds (barium thiocyanate is a product) must be disposed of in an environmentally sound manner. This means they have to be sequestered as chemical waste - not dumped down the sink into our water supplies or thrown into the trash where it can leach into the soil.
If an endothermic reaction cools a flask significantly, water vapor from the air will condense on the surface of the flask. For the unsuspecting chemist, this creates a slippery surface, and can result in an easily dropped (and broken) flask. This occurrence is a boon for demonstrators, however. Reactions that lower the temperature below the freezing point of water can freeze the condensate. The resulting ice is an excellent visual indicator of the temperature change. Taking further advantage of this ability, demonstrators may also set the flask on a wet surface. When the water freezes, the flask cannot easily be lifted from the surface to which it has frozen.
Keeping all these consideration in mind, you should be able to handle any endothermic reaction safely. With experience in the lab, most of this should already be second nature, with only the cold temperatures and condensation presenting unusual risks. Lastly, if you only need a quick demonstration of something that is endothermic, try dissolving some simple salts (sodium nitrite, sodium nitrate, ammonium nitrate, etc). You can obtain a significant temperature drop in these cases, and then recover the chemical afterward, so disposal isn't even an issue. (If you do choose to dump them, they're just salts and can go down the sink anyway.)