tldr; there is a commensurate level of theoretical knowledge that will help immensely with what you want to do. It may not be much theory, but if you know it, you'll be much better off (unless you're a musical genius, as Michael Curtis points out).
When someone with extensive theory knowledge listens to a song, they might think, e.g., "Oh, that's a I-vi-IV-V progression in G." Thanks to their theoretical knowledge, they've recognized the pattern more quickly, they've expended less mental effort, and they'll remember that song for a longer time.
You can probably already see how theory helps. When you play by ear, you're decoding sounds into notes. Theory provides structures and frameworks for conceptualizing both of those things (the sounds and the notes). Theory identifies patterns in music. It's like the teacher standing over your shoulder saying "look at the chords you just played--it's the same sequence as the song you learned two weeks ago." When you spend time thinking about those patterns, learning them, and naming them, you will be faster at identifying them and better at remembering them. Knowing those patterns is essentially a "shortcut" that reduces the required mental effort.
In essence, when you learn a song off the radio instead of reading sheet music, you're transcribing. The less intensive your transcription work, the less theory you'll require. If your desired task is to play melodies and chords, then you don't need to go buy 10 books on theory. Focus on diatonic chord progressions and the major/minor scales, and that will probably be enough.
On the other hand (to use a more extreme example), if you want to transcribe exact chords/notes on a solo jazz piano track, then you'll want to know a wide range of theoretical voicings (upper structure triads, combination triads, quartal voicings, stacked thirds, drop-2, shell voicings, type A/B voicings, etc. etc. etc.).
Theory is nothing more than a collection of statements describing patterns in the music. Knowing those patterns helps. But you only need to study the patterns you care about.