Unfortunately, the $T_i$ are not a poisson process. One way of showing it is, for example, by considering the number $N$ of $T_i$ that falls into the interval $[0,1]$. It's supposed to be a Poisson rv if the $T_i$ are a poisson process. But,
$$\mathbb{P}(N=0)=\mathbb{P}(\tau_1\geq 1)=e^{-\lambda_1}$$
and
$$\mathbb{P}(N=1)=\mathbb{P}(\tau_1\leq 1~\mbox{and}~\tau_1+\tau_2\geq 1)=\left\{\begin{array}{l} \lambda_1 e^{-\lambda_1}~\mbox{if}~\lambda_1=\lambda_2 \\ \frac{\lambda_1 e^{-\lambda_2}}{\lambda_2-\lambda_1}\left(e^{\lambda_2-\lambda_1}-1\right)~\mbox{otherwise.}\end{array}\right.$$
So, $\lambda_1=\lambda_2$ and similarly we can show that all $\lambda_i$ have to be equal.
An instinctive way of seeing that $T_i$ is not a poisson process is imagining, for example, that $\lambda_1=1$ and $\lambda_i=10$ for all $i\geq 2$. If we know that $T_1<1$ then we know that the average number of $T_i$ that falls in $[1,2[$ is $10$ but if $T_1>1$ then this is not the case anymore, it will be lower. So, knowing what happens on the interval $[0,1[$ has an influence on what happens on the interval $[1,2[$. This is never the case for a Poisson point process.