Are weather forecasts such as on weather.com or other apps continuously updated for each location and updated by an operator or is it just a computer automatically updating the forecast?
Your question is extremely broad and it would be comprehensive to answer in detail. How it is done varies from one application to another. But in general, all weather forecasts come from a numerical prediction which is based on an analysis. The analysis is created by combining a prior (the previous forecast) with observations. There are many ways this can be implemented in practice. If you are interested you can start by looking up sequential assimilation and variational assimilation.
Some applications apply a Kalman Filter (a recursive method) to update a forecast when new observations come in, but this is probably becoming less relevant as models start to be run in rapid update cycles, e.g. a new analysis (and forecast) every 10-60 minutes. The weather model itself is a big subject in its own, with numerical approximations, optimisation and parameterisations.
Given the proprietary nature of that information, I don't think that can really be answered.
But my best guess is that it may be updated by a computer, and adjusted by a meteorologist to better align with observations. There is probably some algorithm that mediates between the two.