In true F1 fashion it's not simply called a drink bottle or water bottle, it's called an FDS (Fluid Delivery System) 
found this, not sure if it's still up-to-date:
During the race, a slightly more complex system comes into place, with a medical IV drip bag acting as the drinks bottle, situated in the cockpit to the left hand side of the driver. A system of plastic piping and an electric pump transfer the liquid around the inside of the helmet into the driver's mouth. When he wants a drink he pushes a button on the steering wheel to activate the system and inject the fluid, which can be a simple solution - water.


There is nothing more than a flexible bag of drink attached to the side of the cockpit. To save the driver having to suck the drink up from the bag, it is delivered by a pump. Rather than an expensive titanium-carbon fibre pump, the teams use nothing more extravagant than a road car windscreen washer pump, with the pump linking the fluid bag to the driver’s helmet via a long tube. On the steering wheel, the “drinks” button powers the pump, squirting some of the drink into his mouth. The drivers will call for a squirt of drink most laps when they are on the longer straights.
The drink varies from driver to driver, but usually it’s a high-concentration drink, not a refreshing cool watery drink, mostly made from a glucose-based fluid with vitamins and minerals to boost the immune system and stabilise blood chemistry; much like the sachets of minerals you drink after having a bad stomach. In fact, water would be a bad choice of fluid as it’s not as efficient at replacing the body's fluids as an isotonic drink. Despite the driver needing to keep cool, the drink is not kept cold within the car. The drink soon warms up, and with its sugary and salty taste in the heat, the drink actually resembles warm tea.