Note that datestr outputs a string, while the New_Nun(i,j)=...assignment expects a single value.
datestr(nun(i,j),'HH:MM:SS') outputs a [8 x 1] char array, which will not fit into New_Nun(i,j). And, by the way, 'HH:MM:SS' is indeed the correct argument for hours, minutes and seconds.
You may rather use a cell for New_Nun, which can contain any kind of data including strings of arbitrary length :
%Initialize random values with a NaN
nun=rand([5 5]);
nun(2,5)=NaN;
New_Nun = cell(size(nun));  %Initialize the cell array
not_a_nan = ~isnan(nun);    %Selection valid values
New_Nun(not_a_nan) = cellstr(datestr(nun(not_a_nan),'HH:MM:SS')); %Fill the cell array
disp(New_Nun)    
'13:47:28'    '23:37:19'    '06:00:28'    '13:24:09'    '12:38:05'
'00:55:58'    '08:34:58'    '04:25:44'    '11:14:46'            []
'13:17:25'    '10:34:02'    '11:17:16'    '04:42:18'    '05:51:32'
'20:40:44'    '18:30:16'    '04:29:22'    '17:43:03'    '22:05:24'
'16:36:35'    '10:17:18'    '22:45:41'    '16:04:17'    '07:06:39'
The NaN value has been converted to an empty cell. It could be replaced with any default value, e.g. New_Nun(~not_a_nan)={'00:00:00'};
This code does not handle hours above 24, as datestr. For than, hours need to be treated separately.
%Initialize random values with a NaN and one value larger than 24h
nun=rand([5 5]);
nun(2,5)=NaN;
nun(3,4)=nun(3,4)+10;   % One value above 24h
New_Nun = cell(size(nun));  %Initialize the cell array
not_a_nan = ~isnan(nun);    %Selection valid values
mmss = cellstr(datestr(nun(not_a_nan),'MM:SS')); %Write only MM:SS
hh   = floor(nun(not_a_nan(:))*24); %Get hours, and make it a column vector
%Now, add the hours
hours_not_zero = hh~=0; % Optional: used to display mm:ss if hh=0
mmss(hours_not_zero) = cellfun(@(h,m) sprintf('%02d:%s',h,m),num2cell(hh(hours_not_zero)),mmss(hours_not_zero),'uniformoutput',false);
New_Nun(not_a_nan) = mmss;
disp(New_Nun)
'02:53:42'    '08:38:56'    '12:07:03'    '15:28:54'    '12:49:18'
'16:21:15'    '03:44:23'    '01:22:31'    '20:43'               []
'02:36:04'    '11:08:51'    '19:51:35'    '244:45:48'   '33:55'   
'16:14:08'    '08:04:01'    '12:27:22'    '08:17:48'    '06:16:15'
'10:41:08'    '03:16:19'    '04:47:15'    '04:33:37'    '20:36:42'
This time, the value 35:34:11 is represented correctly. Not significant hours are not displayed, such as '20:43' for '00:20:43'.