What would change in mathematics if we knew $\pi+e$ is rational?

This is a comment that's too long for the usual format. If $\alpha=e+\pi$ can be written as a fraction of integers, both numerator and denominator must be $\geq 2 \times 10^{32}$. To see this, let $A,B,C,D$ be integers defined by

$$ \begin{array}{lcl} A &=& 3063742572717320569341511991159738 \\ B &=& 522834163445445988434458010516405 \\ C &=& 9765222175513935643148512770417523 \\ D &=& 1666455861030599542832067804101203 \\ \end{array} $$

Then any good formal computing system will confirm to you that $\frac{A}{B} < \alpha < \frac{C}{D}$ and $BC-AD=1$. If $\alpha$ is rational, $\alpha=\frac{p}{q}$ with $p,q \in {\mathbb N}_{>0}$, then $u=pB-qA$ and $v=qC-pD$ must be positive integers. But then $p=Cu+Av\geq A+C \geq 2\times 10^{32}$ and similarly $q=Du+Bv\geq B+D \geq 2\times 10^{32}$.

It would imply in particular that $e$ is a period, which it conjecturally isn't. There are deep reasons why $e$ is conjectured not to be a period, stemming (I believe) from Deligne's theory of motivic weights. I recommend taking a look at Kontsevitch and Zagier's fantastic article Periods.