The prompt states:
Let us accept as true that a person can always walk an extra mile. Does the Principle of Induction then prove that a person can walk forever? Where is the fallacy?
No. The Principle of Induction says that a person can walk a finite number of miles $n$ for all $n \in \mathbf{N}$, but does not say that a person can walk infinitely many miles (walking forever).
The fallacy is that Induction may be used to prove that a statement of the form $P(n)$ is true for all $n \in \mathbf{N}$. It cannot be used to show that a statement of the form $P(\infty)$ is true.
There is no fallacy. The use of induction here shows that, under the assumption, if you give the person enough time, any number of miles will be achieved.
Maybe part of the confusion comes from the fact that you talk about induction, but you don't state your proposition $P(n)$ explicitly. To me, here the natural $P(n)$ is $$ P(n)=\mbox{"a person can walk $n$ miles"}. $$ So the induction principle shows that a person can walk any number of miles. To me, this is what it means "to walk forever". Note that if a person "walks forever" at no time the person would have done infinitely many miles.