It is important to note that every estate, regardless of size, must go through probate. Probate is the process through which a court determines how to distribute your deceased loved one's property.
That said, the probate process for smaller estates will likely be expedited. In most jurisdictions, a modest estate can have a faster and less expensive probate process than larger estates.
Lastly, even a modest estate is still an inheritance. Assets left behind by your loved ones should be should be distributed to their heirs. Abandoning those assets means that they will be turned over to the state and put in the Department of Unclaimed Funds.