Generally, does the time it takes to compute (n mod m) depend on the size of n? Example:
Suppose I have a number a with 100 decimal digits, b with 200 decimal digits, and m with 10 decimal digits.

Would the speed of the computation a mod m vary greatly from b mod m because of the size of a vs b?
Yes. You are doing long division. Whatever method you use, you will lose you a roughly constant number of digits per step, so there will be twice as many (well, about $\frac{190}{90}$ times as many) steps for $b$ as for $a$.