Unfortunately 'none' is not a valid value for 'MissingDataColor', and I cannot find any way to set alpha values in heatmaps.
Would something like the following work for you? A unique color (here, pink, but could be anything) is used as the missing data color. Then, after you have finished editing the heatmap however you like, the heatmap is copied to an image. Any pixel in the image whose color equals the unique missing data color is set to transparent using the 'AlphaData' property of an image.
% set up some random data to plot in heatmap
N = 10;
d = randi(10,N);
d(randi(N*N,3,1)) = NaN;
% create heatmap
f = figure;
mC = [255 192 203]; % RGB of color for missing data - should not match any other color in heatmap
h = heatmap(f, d, 'MissingDataColor', mC./255);
% after you are completely done editing heatmap, transfer to image and set image's AlphaData property
c = frame2im(getframe(f)); % color data
a = ones(size(c(:,:,1))); % alpha data
a(c(:,:,1)==mC(1) & c(:,:,2)==mC(2) & c(:,:,3)==mC(3)) = 0; % set alpha data for pixels of missing data to transparent
figure
I = imshow(c);
I.AlphaData = a;
