Would replacing a resistor by another one of identical value but higher power rating affect the working of a circuit?
In many cases the change would be acceptable.

Inductance: In instances where high frequency AC is being dealt with it is reasonably possible that the higher power resistors are wire wound. If special care is not taken to produce a non-inductive resistor a wire wound resistor tends to have inductance due to the spiral of wire that it is made from. This can produce significant effects in some cases - but this is unlikely to be a problem in the vast majority of circuits. Where low to no inductance is required, wire can be doubled and wound onto a former so that current flows from input to halfway point with one rotation and from halfway to end with the other rotation - this cancelling out inductance overall. This places constraints on construction methods and insulation and would be avoided if not needed. 

Voltage rating: For 20 Ohm resistors it will not be a problem, but resistors have voltage ratings and in a high voltage circuit (hundreds of Volts) you need to be sure that a replacement or replacements handle the voltage properly. Generally replacing one resistor with two will make voltage problems less severe.

"Precision": As Pat noted - if the original resistor was high 'accuracy' (say +/-1% tolerance ) and needed the precision, replacing with low precision high tolerance resistors may cause problems. It is not uncommon for power resistors to be +/- 10% rated wheras a 1 Watt resistor is liable to be 5% and maybe 1%.