The ideal cell is modelled by a fixed emf and fixed internal resistance. By definition the emf is independent of both internal and external resistance, temperature, current, etc.
Real cells do not necessarily behave like this. They get depleted, so the emf decreases over time. You have to recharge or replace them. If you draw too much current the emf also gets depleted quickly. The emf arises from a chemical reaction, and if the temperature rises the reaction goes faster so the emf increases.
The emf of a real cell is only fixed and independent of current for times which are short compared with the cell lifetime and for currents which are not too large.