As mentioned in the question's comments by jimsug, the answer to "which one should I use?" is contextual.
Use do not if you're describing the existing state of affairs; if, currently, the contents of a reordered playlist are not displaying in the correct order.
Use will not if you're talking about what will happen in the future; if, when the playlist gets reordered at some later time, the contents are not going to be displayed in the correct order.
Also use will not if you're figuratively ascribing agency to the contents. We often use this construction to indicate a refusal to do something. For example:
Did you ask Jane to file the papers?
Yes. She will not do it.
Even though both the request and refusal occurred in the past, we say will not, because Jane made it clear that at no point in the future will she file the papers.
The use of will not could mean that the contents are actively refusing to appear in the right order. This should not be taken literally, as playlist contents cannot make choices, so obviously they can't refuse to do something. You might hear the will not form from someone quite frustrated that the playlist is not displaying the correct order, and who does not think the problem will be solved any time soon.
I have a colleague who is working on his project that he needs to document, i think, the bugs a system has. One of it is the question I am asking right now, he used "do not", instead of "will not" which I recommend. I know it is more right if he uses will not, but I was not able to explain to him why he has to use will not.
Is this system currently (or will be in the immediate future) used by anyone, or is it still only in development? This makes a difference, because in this situation we are describing the behavior of the system from the perspective of a user (not a developer).
If there are users or testers - people regularly accessing it other than just your colleagues who are making it - then it is a current problem. The contents of any reordered playlists aren't displaying in the correct order at the present time, so use do not. This is particularly appropriate if some user has reported this as a bug.
If only the people making it have access, then you may use will not. This means that when a user reorders a playlist in the future, the contents will not be displayed in the correct order. But, right now, this isn't an issue, because playlists very rarely get reordered, since the people who would do that don't have access.
In the specific context of a bug report, it doesn't really matter which version you use, as both convey the appropriate meaning, though the statements don't carry the same technical semantics.
Do not version: currently, the contents of any reordered playlists are displayed in the wrong order.
Will not version: if you reorder a playlist [the reordering hasn't happened yet], the contents will display in the wrong order.
As pointed out by Damkerng T, your colleague should follow whatever documentation or bug reporting conventions your company uses. If others choose one of do not or will not over the other, your colleague should follow their lead, regardless of whether or not regular users have access to the system.