Are there any phrases expressing the meaning "won't do/finish something until the last minute of deadline/due time"

Solution 1:

Consider to cut it close:

(idiomatic) To judge or finish something close to its limit.

This expression implies that the action was completed on time but suggests that it easily could have been late.

In your sentences it would read:

• Class starts at 8 am. Arrive on time. Don't cut it close.

• You always cut it close with your homework. Why don't you submit it earlier?

Solution 2:

Procrastination, especially in the academic or scholastic contexts of your two examples, is often called

• to cram (vt)
  postpone studying a subject until the few hour prior to a formal examination.
  refers to the “cramming” of preparations in a small available time.

Variantly, the non-idiomatic procrastinator, or to procrastinate, could be adequate.

Solution 3:

They are taking it down to the wire.

down to the wire until the last possible moment The election was so close, it went down to the wire and was decided by a court. We had very little time to get the place decorated for the party, which meant that everyone worked right down to the wire. - The Free Dictionary

Your statements would read:

• The class starts at 8 a.m.; please arrive in the classroom at least 10 minutes in advance. Do not take it down to the wire.

• You always take it down to the wire! Why don't you submit it earlier?