You could put
\DeclareSymbolFont{letters}{OT1}{cmtt}{m}{n}
in the preamble of your document. Though I'm not certain why you would want to use this as the default math font. Also, by making this declaration some symbols get rendered incorrectly:
\documentclass{article}
\usepackage{amsmath,amssymb}
\DeclareSymbolFont{letters}{OT1}{cmtt}{m}{n}
\begin{document}
Hello world:
\[
x^2 + y^2 = \tan(\theta)
\]
\end{document}
which gets rendered as:

There is another approach. But it requires a bit more work.
First, declare a symbol font name
\DeclareFontSubstitution{OT1}{cmtt}{m}{n}
\DeclareSymbolFont{myletters}{OT1}{cmtt}{m}{n}
\DeclareSymbolFontAlphabet{\mathnormal}{myletters}
But this is not sufficient to enact the changes you want. Now you've got to go through for every symbol you want set in this style and declare it:
\DeclareMathSymbol{a}{\mathalpha}{myletters}{`a}
\DeclareMathSymbol{b}{\mathalpha}{myletters}{`b}
\DeclareMathSymbol{c}{\mathalpha}{myletters}{`c}
\DeclareMathSymbol{d}{\mathalpha}{myletters}{`d}
\DeclareMathSymbol{e}{\mathalpha}{myletters}{`e}
etc., and then
\DeclareMathSymbol{A}{\mathalpha}{myletters}{`A}
\DeclareMathSymbol{B}{\mathalpha}{myletters}{`B}
\DeclareMathSymbol{C}{\mathalpha}{myletters}{`C}
\DeclareMathSymbol{D}{\mathalpha}{myletters}{`D}
\DeclareMathSymbol{E}{\mathalpha}{myletters}{`E}
etc. It's a lot of work. You'll be losing a lot work others have already done.
At this point, I find my answer bordering on the questionable. For example, even in the original solution I posted \tan got type-set in the usual default font, and none of the changes I'm suggesting above seem to change that.