LibreOffice v5.4.1.2 (Windows) does not, OpenOffice v4.1.3 neither.
What exactly are you doing?
I do this:
** DRAG AND DROP **
- Cell B3: type value 1
- Cell B5: type vlaue 2
- Cell D3: type formula "=SUM(B3:B5)". ENTER --> value 3 is displayed
- with the mouse I select cells B5 and B6, click somewhere in any of these two cells and drag both cells to a different location
- The sum value in D3 changes to 1. And when I select D3 I still see the old formula
** CUT AND PASTE **
- repeat the first 3 steps of above
- Select cell B5.
- Press CTRL+X --> sum cell value jumps to 1 (because value in B5 is erased)
- Click some other cell
- Press CTRL+V to paste the value back in --> sum cell value remains at 1. The formula in D3 is still SUM(B3:B5)
[EDIT]
if the formula cell C3 contains the expression "=B3+B5" then the behavior is like you describe. But this is even worse: Even within the same application the behavior is inconsistent. Why should "SUM(B3:B5)" behave differently from "B3+B5"?