Errol Garner, the amazing jazz pianist, was once asked why he never learned to read music. "You seen one note, you seen 'em all!" was his rather brilliant reply!
But learning to read is a must for the modern musician. To play with other people, to learn new pieces, to compose (even at a basic level) learning the "dots" makes life whole lot easier.
Don't worry about this, though. It's step-by-step process of learning the notes on the stave (the horizontal lines), time values (notes with and without "tails", blacked in or white), key signatures (different keys - C, D, E, F etc)... It's a whole new and very fascinating world!
We'll advise you on which theory books are best and your teacher will gradually build up your knowledge and confidence.
If you decide, in spite of everything, to just "play by ear" - well there's no law against it!
Note for guitar players - there is a special "tab" system for guitarists, based on the guitar fretboard. An easy way to learn chords.