Thermal drift in diodes
so Io will become smaller That's not true, what you wanted to write is that the term \$(e^{V/V_t}-1)\$ becomes smaller and that is true.

However \$I_0\$ is often / usually (wrongly) assumed to be constant, but it is not. 

\$I_0\$ (also called \$I_s\$) relates to the diffusion current which is VERY temperature dependent. It completely overrules the fact that \$(e^{V/V_t}-1)\$ becomes smaller.