Even clean hands leave behind a residue over time, and your doorbell button is a prime target for accumulations of sticky dirt. Buttons protected by a porch or roof overhang do not get splashed clean by rain and may be more susceptible to sticking. Kids with muddy hands from the yard are also prime suspects when it comes to dirty surfaces.
Use a cloth, some cotton swabs and rubbing alcohol to remove mud and dirt from the surface of the doorbell button. Push the button down and wiggle it from side to side if it is still stuck down. Push the button down a few times to work it in.
Contact the doorbell manufacturer to see if any plastic covers are available for your model of doorbell. Some doorbells come with an optional rubber cover that can easily be removed and wiped clean when it begins to get dirty. The covers also provide protection from weather and water damage.
A small section of plastic kitchen wrap is a useful analog in the absence of a plastic cover. Cut a small piece to size and wrap up the doorbell without applying too much pressure to the button; there should enough inside the plastic for it to move up and down. Change the wrap as it becomes dirty or damaged.
There may be a problem with the button's internal parts if these solutions do not work. Using a screwdriver, take the button apart and separate as many of the pieces as possible. Clean each piece with alcohol, as outlined above, and inspect for damage. Contact the doorbell's manufacturer if you need a replacement button.