Can you measure the temperature of a flame of a Bunsen burner using a thermometer?
Yes, and No.
You will get an inaccurate number since of air temperature and a few variables, but it will be relatively close.
I do not recommend this, because a few things may happen, and even some I do not even know of. The thermometer will get hot and melt. Or explode. And many other things can happen.
The temperature of a Bunsen burner is regulated by gas and airflow. The gas is controlled by a flow valve and the air is controlled by a screw mechanism on the collar. Different type of nozzles can control the flame's shape. The Bunsen burner was invented in 1855 by Robert Wilhelm Bunsen (1811-1899).
Place the thermometer in the liquid you are measuring. Wait until the temperature stabilizes, and then read the thermometer at eye level, just like you would a graduated cylinder, except that there's no meniscus. If you are measuring a liquid that is being heated over a Bunsen burner, you should have the thermometer held in place with a thermometer clamp. You should have the thermometer positioned about halfway down in the liquid.
Heat it. Place a 300ml sample of the cold water in a 500ml beaker and place the beaker on a retort stand about six inches above a standard laboratory Bunsen burner. Turn on the gas and ignite the Bunsen burner. Allow the flame to heat the water for several minutes. When small bubbles begin to form in the water, shut off the Bunsen burner and measure the temperature.