The most common and most likely answer to this crossword clue is the 7 letter word INFARCT
I've seen this crossword clue in The Guardian
I
N
F
A
R
C
T
More clues leading to the result INFARCT
Buddy explains!

The straight part
We know that "Dead heart, say" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case dead heart, say = infarct.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does INFARCT mean?
INFARCT noun- An area of dead tissue caused by a loss of blood supply; a localized necrosis.