If the person who died was paying tax on income from investments or as a self employed person or as an employee, tell the tax office about the death as soon as possible. This will enable the deceased's tax affairs to be settled. Depending on circumstances, this may involve some more tax or claiming a repayment.
The particular tax office to contact will depend upon the deceased's circumstances.
For instance:
Inland Revenue leaflet IR45 What to do about tax when someone dies gives more information. Alternatively, further information, and copies of IR45 leaflet are available at the Inland Revenue Website