Can anyone tell me what parameters affect the proper functioning of a micro-controller based circuit, if we were to keep the power ON for days together?

The law of probability!

If you design an electronic circuit, then you take into account a certain level of interference. Your design measures are choosen in order to make your circuitry immune against this level of interference. But how to decide about this level of interference? It's a compromise between total protection and costs. So, in most cases a level is choosen, which is identical to some standard, like FCC rules or CE laws. But, you don't have any guarantee, that this level of interference is never exceeded. It's that, what you can assume by applying the 'common sense'.

With a 'normal' product, which is switched-on only for some hours a day, all the time benefitting from the presence of operator, very unprobable conditions are normally not taken into account. Because, if the product fails for some reason, operator uses well known power on reset feature...
Such a product is called 'toy'.

If on the other hand a 'forever' application is designed, these very unprobable conditions must be taken into account. Why? Because the very low probability accumulates over the time and becomes a serious problem. So, for a 'forever' product you must take into account a level of interference which is much higher than that assumed in the FCC rules or CE laws. But even then, if you take into account an 'idiotic' level of interference, you will never have the guarantee, that not even bigger interference will hit your system, one fine day.

So, your application is always ranking between the 'toy' and the 'real forever' design. Towards which side depends on you, on your willing to invest money, volume and developement time for filtering parts, protection circuitry, shielding, grounding, isolation, etc. etc. etc.

Make no mistake: You cannot design a system which runs forever! And "Can my system run forever?" is not the right question, but more "Does it hurt, if the system fails?". Another good question is "How to find out, whether the product has a failure?"

Increasing reliability is one of the most demanding challenges of the next future. Only if we can make electronics much much more reliable than today, we can increase the amount of electronics in our live, without running into danger. Think of electronics in cars, aeroplanes, nuclear power stations, space shuttles, manned Mars stations, etc.

Kai