Let $T\geq 0$ and let $(X_i)_i$ be a sequence of iid non negative random variables such that $E[X_1]>0$.
Let $\tau = \max\{n\geq 1|\sum_{i=1}^n X_i\leq T\}$.
How could I upper bound $E[T-\sum_{i=1}^\tau X_i] = T - E[\tau]E[X_1]$ ?
Is it true that $ \frac{E[\tau]}{E[X]}= \lfloor T\rfloor$?
This is an answer to the second part, and an argument that there should not be a nice answer to the first part; indeed the equality you state is incorrect. We need not have $E[T - \sum_{i = 1}^\tau X_i] = T - E[\tau]E[X]$ since $\tau$ is not a stopping time.
Let $Z_n$ be the random variable with $P(Z_n = 1/n) = (n-1)/n$ and $P(Z_n = n) = 1/n$. Then as $n \to \infty$ we have $E[Z_n] \sim 1$.
Let $\tau_n$ denote the corresponding $\tau$ variable; i.e., take $X_j$ to be i.i.d. copies of $Z_n$ and set $\tau_n = \max\{m : \sum_{j = 1}^m X_j \leq T\}.$ For any fixed $T$---e.g. $T = 1$---we will have $E[\tau_n] \to \infty$ as $n \to \infty$. This also shows that the equality you state is not correct, since this demonstrates that the right-hand-side may be negative even though the left-hand-side is non-negative.