This is because the iteration variable in Sum is treated as local using Block. You can use With to replace all occurrences of a symbol inside an expression with a value (even inside held expressions) So

With[{r = Extract[{0, 0, x, y, 0}, 3], 
    inf = Extract[{0, 0, x, y, 0}, 5], 
    sp = x + y}, 
    Sum[t^sp, {r, inf, Infinity}, 
      Assumptions -> {t < 1 && t > 0}]]

will give what you expect.

$\frac{t^y}{(1 - t)}$

If I understand your question correctly, you're asking how to replace the iteration variable in sum with some other selected variable.

The reason your code doesn't work is that, in Sum the iteration variable is localised by making use of Block. You can use With to replace all occurrences of a symbol inside an expression with another value (even inside held expressions) So

With[{r = Extract[{0, 0, x, y, 0}, 3], 
    inf = Extract[{0, 0, x, y, 0}, 5], 
    sp = x + y}, 
    Sum[t^sp, {r, inf, Infinity}, 
      Assumptions -> {t < 1 && t > 0}]]

will give what you expect.

$\frac{t^y}{(1 - t)}$