The forward voltage of a diode will drop with junction temperature, at typical currents and around room temperature, by about 2mv/K or -0.3%/°C.
At high temperatures the reverse leakage will be much higher and the forward drop will be somewhat less.
If that difference matters to you (either because you expect junction temperature to be far different from room temperature- due to self-heating or ambient variations or both- or because you need to model fine changes), don't use the approximation for Vt. It's more than adequate for many purposes, totally inadequate for others, and marginal at for others.
The mathematical model you use for a given component is a trade-off between accuracy and simplicity. For example, real diodes have an ideality factor \$\eta\$ as well as a resistive component to the behavior, both of which can be very significant. Also Is in the Shockley equation is temperature-dependent.