They're both absolutely correct! Here's a concise version of what they said: "I will do the follow up" is fine, since you already have a verb "do" and the "follow up" is a noun. "I will follow up" is also correct, as "follow up" in this case is a verb.

I will follow up. This means that having done something (that perhaps needs more investigation before being completed) YOU will continue the investigation, on your own, and report back (to your classmates, the research team, to the boss).... but NOT to the teacher since you are expected to provide completely researched answers by the time the homework was asked for. It is also used in the context of "TO FOLLOW UP" where you meet someone and will contact them again either with more information or to continue a conversation. Sales persons often say this when you are looking for a product or a service "I will follow up with you ..tomorrow, this afternoon...) Finally, Follow up can be used medically as in "A FOLLOW UP EXAMINATION. So after a surgery, the doctor says "I will set up a follow up appointment" to remove the sutures".

"follow up" is not an easy phrase. It can be used as either a verb or a noun. If it is a verb, it means "finish what has been started", and your sentence becomes "I will follow up". If it is a noun, it means "what we have not finished yet", and your sentence is correct. Like I said, it is a bit complicated, but that should explain most things. Hope that helps!