Does it damage my piano if it’s not tuned often enough?

Usually not. But, if a piano has not been tuned regularly, it may need what is called a “pitch raise” or a “pitch correction”.
This is a procedure to

 properly and safely bring the piano’s string tension back to normal. Over the years, Piano Clinic has developed a process to do this more quickly, and therefore, for a lower fee than many other tuners will charge. But keep in mind, after a pitch correction, your piano may need one or more additional tunings over a period of months for it to stabilize.