Doing the addition as is, we get $641 + 852 + 973 = 2466$. This number is $10$ larger than the desired sum of $2456$. Therefore, we must make one of the three numbers $10$ smaller.
We may either change $641 \rightarrow 631$, $852 \rightarrow 842$, or $973 \rightarrow 963$. Either change results in a valid sum. The largest digit that could be changed is thus the $7$ in the number $973$, and the answer is $\boxed{\textbf{(D) }7}$.