Is it reasonable to update the technical specification every time changes are made during the development of the project?
Yes, it's worth the time to update the specs.
Technical specification ≠ documentation, but often there is no documentation at all.
So if changes are made during the course of long project there is no reference if the specs aren't kept up to date. After a while there's a lot of guessing and the only way to find out is to ask engineers to look up the code (which is time consuming and makes them less productive). Or, if there's a system already live and working, it requires running time intense a test on this system to find out about the behaviour.
If the technical specification and/or documentation is up to date there is no guessing and it makes everyones life easier. Plus, it makes your company look more professional because you have always a quick and correct answer to the customers' questions.
And in an ideal world the technical specification is updated before the change was implemented in the code.