Both more and else are syntactically fine in OP's example, and in many contexts they'll mean exactly the same thing.
But note that idiomatically, What more do you need? is far more likely when what's being asked is effectively a rhetorical question (implying the speaker thinks you either don't or shouldn't need anything else).
Also note that when using else , to some extent the distinction between a rhetorical question and a genuine enquiry can be made more explicit by stress/emphasis... 1:
What else do you need?
(Probably: You don't need anything else - just get on with it) 2:
What else do you need ?
(Probably: If you need anything else please tell me)
ell.stackexchange.com/users/126/FumbleFingers Reinstate Monica
In the context you have given, can mean different things.
What more do you need? Can be taken as for example -
"Isn't what is given enough?
What else possibly could you need" implying that what is given should suffice.
What else do you need? Can be taken as for example -
"What are the other necessities that I can provide to you?
What other requirements do you have?"
ell.stackexchange.com/users/16013/user1860175
You can say : What more do you need?
(When you are really vexed up with that one)
You can say : What else do you need?
(when you were really happy to help him, and you were asking do you need something more(Soft Tone) )
As far as I known.
ell.stackexchange.com/users/None/
They both work but are slightly different.
else means;
In addition, besides, or different and instead.
What else (different, besides, or instead of) do you want?
more means; A greater or additional amount or degree.
What more ( A greater amount, or in addition to ) do you want?
So the main difference is that else , in general means different .
And more , generally means A greater amount or In addition to .
ell.stackexchange.com/users/18828/Sam Harrington
Both are acceptable but my personal opinion when you say "what more do you need" it sounds as if you provided more help than they need so there is a connotation that you think they don't need something anymore whereas "what else do you need" sounds neutral that you don't know their needs even if you provided some help before.
ell.stackexchange.com/users/11631/Mrt
I am providing you with food, shelter, clothes.    What more do you need   or    what else do you need?   Which one's correct?
"What more" vs "what else" do you need?
ell.stackexchange.com
<sentence-usage>
ell.stackexchange.com/questions/87952
ell.stackexchange.com/users/15446/Ardis Ell
Both  more  and  else  are syntactically fine in the given example, and in many contexts they'll mean  exactly  the same thing.  However, they are slightly different. You should note that idiomatically,  "What  more  do you need?"  is far more likely when what's being asked is effectively a  rhetorical question  (implying the speaker thinks you either don't or  shouldn't  need anything else).  Also note that when using  "else" , to some extent the distinction between a rhetorical question and a genuine enquiry can be made more explicit by stress/emphasis... Both questions have correct syntax and can be used interchangeably to mean the same thing. However, often they  have slighly different meanings. While both  are acceptable, "what more do you need" is more likely to be a rhetorical question and there is the connotation that the speaker feels they have provided more than enough help and that the other person should't need anything else, whereas "what else do you need" sounds more neutral. Also, "what else do you need" can sound more like a genuine enquiry by adjusting the tone of voice.
Both questions have correct syntax and can be used interchangeably to mean the same thing. They often have slighly different meanings. While both  are acceptable, "what more do you need" is more likely to be a rhetorical question and there is the connotation that the speaker feels they have provided more than enough help and that the other person should't need anything else, whereas "what else do you need" sounds more neutral. Also, "what else do you need" can sound more like a genuine enquiry by adjusting the tone of voice.