Does it matter that your dissertation has been submitted when you apply to a postdoc position? I'm finishing my fourth year of a Ph.D. in mathematics. I have enough for a dissertation, and by the end of the summer, I will probably have enough for a very nice dissertation. And I plan to finish next year.

I also a couple of published papers not directly related to my thesis, and a few projects that might bear fruits maybe even this summer. So by October, when "application season" begins, I should be ready to apply to schools.

My advisor wants me to submit my work around the time I apply, so I can write in the applications that the thesis has been submitted. However, if I do that, I am likely to find myself without funding and without money for a significant portion of the year, as my university halts your stipend and employment as soon as your work is approved.

His argument is that a lot of universities might outright reject, or at least consider it less favorable, if I haven't submitted my work. As it might signal that I'm not ready to move on. This seems a bit odd to me. But I never sat in the chairs of those who make these decisions.

I also don't know how these things vary between departments, so I'm asking specifically for mathematics.
This answer is specific to the USA.

Your dissertation advisor needs to write in their letter something along the lines that you've written X chapters of your dissertation (or the equivalent in the hard sciences) and that they fully expect you will be able to defend and submit by such-and-such date. 

On the receiving side, we know that some students will decide at the last minute to not submit their dissertation and defer graduation. This is usually critically important for international students who will lose their immigration status once they graduate, but it also applies to domestic students as well who might need university health care, housing, or just the extra time to perfect their dissertation.

We all know how this game is played. On the receiving side, all we need is assurance that you WILL be able to submit IF you are given the post-doc. Many post-docs have been burned by recipients who did not finish the dissertation receive the PhD before arriving at the post-doc and turned the post-doc into a pre-doc. Many now have formal language that if you do not submit by the start of the post-doc, your hiring letter will be vacated.