It's subjective. Some people just seem to have inbuilt good timing, others, who've played for years, don't seem to notice a skipped beat, an added beat, slowing down, etc. And for those, a metronome probably won't help. I've tried drum machines with them, and they didn't help either.Generally they're folks that play by themselves for long periods.
It depends a lot on what you're doing with a song. If you're working through the chord sequence, strumming along, then the tempo of a track should be enough to keep you going.If the changes are too quick, then by all means slow it down, but it's probably only one bit that's tricky. Just go over that part slowly.
If it's a solo, and you're trying to read tab,(I know you said by ear), it'll have to be slow, but just do it at your own speed till it's good enough.With solos, try to understand the sets of notes, and align them to notes you've learned in scales, etc. Patterns will appear, and they will help the flow of your playing.And for Gawd's sake, don't rely too heavily on tab, especially that with no proper dots for timing ! A good way is to loop a phrase and keep listening and playing along to it - try to loop it to exact bars, though.
I've mixed views on metronomes, and click tracks. It's so much better playing along to a drummer (especially a good'un), or a drum track. A metronome to me is quite un-musical.Use a backing track when possible.Ask your teacher to elaborate on why a metronome is the 'best way to learn correct timing...' For some, a tapping foot is the best. You'll know you've got it sorted when you play along with others, and no-one moans !