This will extend the life of the tankless.
It will indeed extend the life of the water heater, because the filter will capture and trap any debris or particles that would otherwise interfere with the operation of your tankless water heater.
How crucial is this?
What it will not do is prevent or replace what is crucial, and that is the maintenance of the water heater.
An ongoing maintenance requirement is to make sure your heat exchange elements are free from scale buildup, usually lime scale. What looks like a light coating of scale buildup can have serious impacts on the efficiency of the system. If left unchecked, such buildup can cause longer burner cycles, leading to a strained system and a shorter lifespan.
We do not have soft water here in Los Angeles. Does this factor into the decision of whether or not to install a filter?
I have a water well and the water is definitely hard and a water softener has been installed. If the water softener levels are off a little bit, the lime/scale and calcium build is much faster. Shower heads get clogged etc. This is the same thing taking place in the water heater. So again maintenance is important and do not solely rely on the filter by any means.
If you have the finances, I would strongly suggest the water softener.