I contacted MATLAB technical support. Here's the response I got:
In manual placement mode, the text labels are text objects that are placed on top of the contour object. This gives users full control and flexibility for the label placement, but the labels are now managed independently from the contour, and the contour object doesn't have the information about the labels. Therefore, the contour object will not add gaps to the lines to accommodate the text labels.
If you want to have the space for labels, a workaround is to add a white background color to label text objects, which will obscure the contour lines and make it looks like a gap in the line. But this solution is not ideal, as it will overlap anything (including the fill or other graphics and lines in the contour), but it could be a possible workaround for your workflow.