Leaving the capo on while playing the guitar will not damage your strings but you shouldn’t leave the capo on the fretboard when you’re not using the instrument. It may not damage the strings visibly but it will certainly wear them out which in time will lead to damage if you continue to leave the capo on all the time.
When clamped on the neck, the capo will hold the strings down and create extra tension on the top and neck of the guitar. While all guitars are destined to have problems at some point in time due to the tension of the strings, using a capo all the time can hasten the process.
Instead, you should remove the capo off your fretboard after you are done playing the instrument. This shouldn’t be a problem since most of the products currently on the market are designed to be stored behind the nut when not in use.