Tuning a piano once a year is a minimum, twice a year is optimal.
Seasonal humidity changes are the leading cause of pianos going out of tune. Excessive dryness can cause your piano to develop problems like loose tuning pins, cracked soundboards, or rattles. Too much moisture causes sticking keys, sluggish action and rust. Some people opt to have a Piano Life Saver system installed to neutralize this effect and help stabilize the tuning over longer periods (see humidity control systems).
Tuning of neglected pianos can be more challenging (and costly) so it is better to tune regularly.
Yes! Tuning your piano on a regular basis can save you money in the long run. Pianos which go for long periods of time without being tuned will still drop in pitch (go flat). This then requires a "pitch raise" at the next tuning, meaning it will have to be tuned twice - once to bring the strings close to pitch and another to fine tune. In extreme cases it may require a third tuning to stabilize the tuning. Also, letting a piano drop significantly below pitch increases the chance that a string might break when it is tuned. Ultimately this can be more costly than maintaining your piano on a regular basis.
The A string above middle C is adjusted to vibrate 440 times per second or 440 Hertz. This is the recognized standard around the world for tuning. Most pianos have around 230 strings and each string pulls between 150-200 lbs. of pressure, depending on the length and thickness of the string. The combined pressure of all the strings on a medium size piano is around 18 tons (a concert grand around 30 tons)! A manufacturer's warranty often requires yearly tuning to maintain the pressure between the soundboard, the bridges, the harp (the cast iron plate) and the wooden frame to the specifications the piano was designed.