When building a circuit to power an LED, we use Ohms law to calculate the required resistance, then to calculate the required wattage of the resistor. Suppose this formula dictates a 1/8 W resistor, and I instead use a 1W or a 100W resistor of the correct resistance. What will happen?
In your example, you've paid too much money and the part is huge (especially the 100W part). If the LED current is an AC signal (PWM or whatever) the 100W resistor may have a lot of inductance, which will change the way the circuit behaves.
On the other hand, the ratings of resistors are conditional on a bunch of things, such as mounting, surrounding PCB patterns (especially SMT resistors) and, of course, ambient temperature. The number in the part description is just sort of a rough guide, you really have to drill down using datasheets and part series manuals and other manufacturer's data to get the details. You should also consider odd conditions such as extremes of input voltage, what happens if an LED shorts out, and so on.
It is also not a good idea to get too close to the maximum ratings if you want long life. Using a 1/4W part for 1/8W actual dissipation is not a bad idea. Using a 1W part might make sense if it's a high-reliability design or you have a bunch of them on hand. Using a 100W part is silly.
One major issue, aside from the minor increase in capacitance concomitant with the larger size is that many large power resistors are wire wound. This often means they also function very effectively as power inductors.
Internally, many power resistors are resistance wire wound around a ceramic form, which means they're basically an air-core inductor.
If you are using such a resistor in a current-sensing application in a switch-mode circuit, you will get spurious readings or inaccurate behaviour.
While not all power resistors, or even all ceramic power resistors are highly inductive (see Ayrton-Perry windings), some are very inductive, and it's something you need to be careful of.
Using higher power resistors than necessary is not a problem, other than the additional size and expense.
I see you are considering getting a selection of 1 watt resistors to have on hand. May I suggest 1/4 watt instead? 1 watt resistors have heavy leads that won't fit in the popular plastic breadboards, while 1/4 watt leads will fit easily. In my experience, 1/4 watt resistors are fine for 99% of applications, and can be connected in series and/or parallel if somewhat higher power is needed.