Is there bias when you estimate a truncated distribution in this way? Let's say I have a process which generates non-negative integers based on some unknown probability distribution, however those numbers can not be more than some threshold $N$ (by the design of the system).  For example, let's say the threshold is $N=1000$. This system can generated a number from set $\{0,1,2,4,5,6,7,8,9,10,...,1000\}$ with unknown probability distribution.
Now, suppose I assume a statistical distribution to model the probability of above result which can take any value in positive domain --- e.g. the Poisson distribution, which theoretically can assume any non-negative integer.
Suppose I sample numbers from this process and get $ X_1, X_2,..., X_{50}$.
Now to calculate the expectation, say I use these values and corresponding probabilities from the Poisson distribution.  In this case, would I under-estimate or over-estimate the true mean of the process?
I guess I will be under-estimating the true mean because given that to represent probability I am assuming all positive values can occur with certain non zero probabilities, which is not the case for this system. Therefore I am assigning lesser probability to the corresponding number to estimate the expectation.
Is above thought process right? Will it be the case for any value of the threshold?

It depends on the estimation method you use, which you haven't really specified clearly.  From what you have written here, I take it that you are concerned with the case where you estimate an unbounded distribution first and then truncate the result, instead of estimating directly from a truncated distribution.  In that case, you might over- or under-estimate the true mean of the distribution, depending on the particulars of the distribution and the estimation process.
$$\begin{matrix}
\text{Method 1} & & \text{Estimate non-truncated distribution} & & \text{...now truncate}, \\[6pt]
\text{Method 2} & & \text{Truncate the distribution} \quad \quad \quad \quad \ \ \ & & \text{...now estimate}. \\[6pt]
\end{matrix}$$
You seem to be describing the first method above, where you take an initially non-truncated distribution and you estimate it first, prior to truncation.  You are right that, on the one hand, your initial estimate will ascribe non-zero probabilities to values above the threshhold, which would tend towards over-estimating the true mean.  However, you have not taken account of the fact that your data will only ever fall below the threshold, so this will operate in the other direction.
If you were to use method-of-moments estimation using the Poisson distribution then the first method will consistently under-estimate the mean (see below).  However, if you were to use maximum-likelihood estimation then things are more ambiguous.

An example: To run with your example, suppose we use a Poisson distribution (which is not truncated) with rate parameter $\lambda$ and we estimate the non-truncated distribution using the method-of-moments.  As a preliminary result, define the partial exponential function by:
$$E_N(\lambda) \equiv \sum_{k=0}^N \frac{\lambda^k}{k!}.$$
For a truncated Poisson distribution with upper-bound $N$ we have mean:
$$\begin{align}
\mathbb{E}(K)
&= \frac{\sum_{k=0}^N k \cdot \text{Pois}(k|\lambda)}{\sum_{k=0}^N \text{Pois}(k|\lambda)} \\[6pt]
&= \frac{\sum_{k=0}^N k \cdot \frac{\lambda^k e^{-\lambda}}{k!}}{\sum_{k=0}^N \frac{\lambda^k e^{-\lambda}}{k!}} \\[6pt]
&= \frac{\sum_{k=0}^N k \cdot \frac{\lambda^k}{k!}}{\sum_{k=0}^N \frac{\lambda^k}{k!}} \\[6pt]
&= \frac{\lambda \sum_{k=0}^{N-1} \frac{\lambda^k}{k!}}{\sum_{k=0}^N \frac{\lambda^k}{k!}} \\[6pt]
&= \frac{\lambda E_{N-1}(\lambda)}{E_N(\lambda)}. \\[6pt]
\end{align}$$
Under the first method we estimate $\hat{\lambda} = \bar{x}_n$ for a non-truncated Poisson distribution and then we truncate, which gives the estimated mean:
$$\widehat{\mathbb{E}(K)} = \frac{\bar{x}_n E_{N-1}(\bar{x}_n)}{E_N(\bar{x}_n)}.$$
Under the second method we estimate directly from the truncated Poisson distribution, which gives:
$$\widehat{\mathbb{E}(K)} = \bar{x}_n.$$
The second estimator is an unbiased estimator for IID data, and in this example the first estimator is lower than the second, so it will tend to under-estimate the true mean of the truncated Poisson distribution (i.e., it has negative bias).
