In an engine-out scenario, the remaining engine(s) are operated at a higher power. In itself, this does not require additional checks, just like if the crew was required to make a max-power takeoff versus a lower one.
However, this additional stress may push the remaining engines over operating limits which requires maintenance action. The most typical sign would be engine exhaust gas temperature (EGT) over limits. Typically as an engine ages and it becomes less efficient, it requires more fuel and moves less air, increasing EGT. High EGT is a reliable indicator of wear, and obviously can cause damage itself, e.g. melted parts. The two regimes where engines first hit EGT limits are operation at high powers, and starting.
If an EGT exceedance occurs, this is logged by the avionics and mechanics will action a procedure. Depending on the magnitude (degrees and time), this may mean no action but watching the engine, a borescope inspection, or taking the engine off the wing for overhaul.
Other common exceedances are overthrusting/overboosting, when the primary thrust setting parameter exceeds limits (N1 rotor overspeed, EPR too high). This is rare in modern computer-controlled FADEC engines, as the control loops manage this well, but occurs in older engines. Excess vibration also is a sign of damage, pointing to some physical change internally.