Lyric discussion by Mrs. Lovett 

Cover art for A Little Priest lyrics by Sweeney Todd: The Demon Barber of Fleet Street

What does she mean when she says "No, you see the trouble with poet is how do you know it's deceased?"

Actually, I always thought it was a take on the idea that the work of artists, especially poets, live on well after they are deceased. So in essence, the work of a poet is not forgotten because they are immortalized by their art. So if you apply it to the circumstances in Sweeney Todd, a prominent artist or poet would likely be missed and their disappearance suspicious. That's how I view this particular line anyway.