You can still estimate parameters by using the likelihood directly. Let the observations be $x_1, \dots, x_n$ with the exponential distribution with rate $\lambda>0$ and unknown.
The density function is $f(x;\lambda)= \lambda e^{-\lambda x}$, cumulative distribution function $F(x;\lambda)=1-e^{-\lambda x}$ and tail function $G(x;\lambda)=1-F(x;\lambda) = e^{-\lambda x}$. Assume the first $r$ observations are fully observed, while for $x_{r+1}, \dots, x_n$ we only know that $x_j > t_j$ for some known positive constants $t_j$. As always, the likelihood is the "probability of the observed data", for the censored observations, that is given by $P(X_j > t_j) = G(t_j;\lambda)$, so the full likelihood function is
$$
L(\lambda) = \prod_{i=1}^r f(x_i;\lambda) \cdot \prod_{i=r+1}^n G(t_j;\lambda)
$$
The loglikelihood function then becomes
$$
l(\lambda) = r\log\lambda -\lambda(x_1+\dots+x_r+t_{r+1}+\dots+ t_n)
$$
which has the same form as the loglikelihood for the usual, fully observed case, except from the first term $r\log\lambda$ in place of $n\log\lambda$. Writing $T$ for the mean of observations and censoring times, the maximum likelihood estimator of $\lambda$ becomes $\hat{\lambda}=\frac{r}{nT}$, which you yourself can compare with the fully observed case.
EDIT
To try to answer the question in comments: If all observations were censored, that is, we did not wait long enough to observe any event (death), what can we do? In that case, $r=0$, so the loglikelihood becomes
$$
l(\lambda) = -nT \lambda
$$
that is, it is linear decreasing in $\lambda$. So the maximum must be for $\lambda=0$! But, zero is not a valid value for the rate parameter $\lambda$ since it do not correspond to any exponential distribution. We must conclude that in this case the maximum likelihood estimator do not exist! Maybe one could try to construct some sort of confidence interval for $\lambda$ based on that loglikelihood function? For that, look below.
But, in any case, the real conclusion from the data in that case is that we should wait more time until we get some events ...
Here is how we can construct a (one-sided) confidence interval for $\lambda$ in case all observations get censored. The likelihood function in that case is $e^{-\lambda n T}$, which has the same form as the likelihood function from a binomial experiment where we got all successes, which is $p^n$ (see also Confidence interval around binomial estimate of 0 or 1). In that case we want a one-sided confidence interval for $p$ of the form $[\underset{\bar{}}{p}, 1]$. Then we get an interval for $\lambda$ by solving $\log p = -\lambda T$.
We get the confidence interval for $p$ by solving
$$
P(X=n) = p^n \ge 0.95 ~~~~\text{(say)}
$$
so that $ n\log p \ge \log 0.95 $. This give finally the confidence interval for $\lambda$:
$$
\lambda \le \frac{-\log 0.95}{n T}.
$$
