I keep being told that I'm singing an octave lower than what the music is written at.

For Example, I was singing at D4, instead of D5. I was told this is bad, and I'd be better off singing a few notes lower, rather than a whole octave.

Is this true? If so, do I just need to improve my listening skills, and strengthen my head voice? I'm a tenor by the way.

D5 would be high for a tenor! A tenor's typical range is C2-C4. 

Are you asking about singing down the octave in a vocal warmup or in a song? 

In a song. Yes, I can't hit D5!

May I ask which song? I can help you choose a good key for tenor range!

Hero, by Mariah Carey. But I've now found a song I'm more comfortable with. Boyz ii Men, it s so hard to say goodbye.

Maybe I should stick with a song that's more in my range?

That's a good plan!

If you do want to stick with Hero, there are a handful of "male key" tracks here: https://www.youtube.com/results?search_query=hero+mariah+carey+male+key.