No, the 125degC hotplate won't damage any components (not even most temperature-sensitive components like plastic connectors, electrolytic capacitors etc.) even if you work for dozens on hours on it. Your hotplate temperature is spot on and reduces the risk of soldering issues; also reducing thermal shock to some component types that may be sensitive to quick temperature changes (such as large MLCCs).
125degC is also low enough so that bringing anything on the top of the plate surely won't cause rapid thermal cycling. It would be a different thing to repeatedly bring boards on a 300degC hotplate, for example, due to the sudden thermal shock every time.
Having power applied while hot may be a problem, though, if the junction temperature is close to 125 degC, especially if there is some power dissipation in the components themselves; the circuits won't be designed for operating at Tambient=125 degC. Especially if the thermal design of the circuit is marginal, I would let the board cool down for a few minutes before applying power.