Answer: C.
Solution: (1) When $x \leqslant -2$, the original expression is $=-x-1-x-2-x+3=-3x$. When $x=-2$, the minimum value is 6.
(2) When $-2 < x \leqslant 3$, the original expression is $=x+1+x+2-x+3=2x+6$. When $x=-2$, the minimum value is 2.
(3) When $x > 3$, the original expression is $=x+1+x+2+x-3=3x>9$, so the minimum value is 5.

Therefore, the correct answer is C.
6. The minimum value of $|x+1|+|x+2|+|x-3|$ is ( ).
(A). 3
(B). 4
(C). 5
(D). 6